In general, I think the best motivational basis for study towards state of the art algebraic geometry (or any other subject in science for that matter) is the historical route, and by this I mean to learn the subject step by step since ancient times through the questions and problems mathematicians tried to solve from the Greeks until now, instead of through standard curricula. This may sound challenging but it is not meant as a real historical study, i.e. with the original sources and such, it is intended to mean learning the subject as motivated gradually from the initial problems people faced. Despite of using modern techniques and notation, that is more or less the pathway walked by physics students: we start from elementary phenomenological laws and concrete problems, which then are put together under the mechanical laws of Newton and all its applications (this is the typical first course in Elementary Classical Mechanics), then as one comes across harder physical systems the methods of Lagrange and Hamilton are introduced, thus motivating concepts like energy and more sophisticated theoretical techniques, for example in ordinary differential equations and calculus of variations (this is the typical advanced course in Theoretical/Analytical Mechanics). This is exactly as it happened historically. This kind of historical development of the whole subject, guided by the problems we try to solve, is the source for discoveries and new understandings (for example that conservation laws are actually consequence of symmetries, by Noether's theorem). In physics this gradual learning is a requirement if one is to understand how the discipline is built upon itself: there is no point in trying to grasp quantum field theory without quantum mechanics and this without classical mechanics.
In mathematics, on the contrary, most texts (above all graduate level) build the theory from scratch in a formal unmotivated style, usually without pictures, leaving behind all the motivational and historical background and without mentioning the insights, structures and problems that led us there and whose understanding is the final purpose. Personally, I think extreme logical rigor is the core of mathematics and mere heuristics should never be a substitute, but I never understood why both cannot go in parallel. For that I believe, that besides of the typical numbered points of definitions, propositions, lemmas, theorems, corollarys and their proofs, there should be numbered motivation points: in the same sense that nobody expects theorems without proper previous definitions, a student should not expect definitions and examples without motivations (these are usually "left for the reader [to develop his/her own examples and pictures]" or briefly mentioned in footnotes or small remarks). (As a side note, I personally think that a special mark should be used to point out which propositions or theorems should be done by the reader as exercises, but their proofs be included in the main text, thus avoiding the unnatural splitting of results and the self-learner's nightmare of problems without hints/solutions). Thus, when trying to study advanced mathematics like algebraic geometry one is faced with the discouraging problem of "what is the point of all this?". This is why the historically-motivated learning path is probably the best approach, above all if one is self-studying, so that one can see how problems and solutions come up naturally in a fluent way, which does not mean that their discovery was easy!.
Attempt at the initial question: with enough discipline and hard work, any intelligent person may be able to self-study any subject, or at least get as far as possible. Having a very modest mathematical background is just a problem of needing more time to work through all the requirements: in high school many of us have a modest background in everything but nevertheless end up getting a degree in something. Self-learning is not much harder once you get to it, and in fact is much better for actually understanding anything at all! This is because you are forced to self-teach you any concept, example and problem so you have to fight fiercely and on your own against it all, look up many references and build your own picture of the subject. A coherent path for learning algebraic geometry from a high school background is not so much different that actually going through the core courses of a mathematics degree, one can try to focus on algebra and geometry, but the more mathematics one knows the better to understand all the interconnections (this is just my personal choice, there are lots of other possible references):
Learn Classical Geometry: try to see why ancient cultures needed to measure and what results they could prove: Geometry: Our Cultural Heritage by A. Holme; and learn the basic and visual geometry that dominated for centuries and how its algebraic formulation evolved up to analytic geometry and primitive algebraic geometry: Geometry by Its History by A. Ostermann and Geometry by R. Fenn, or Elementary Euclidean Geometry: An Undergraduate Introduction by C.G. Gibson. For a detailed account of the evolution of geometry and its main topics there is the masterpiece: A History of Geometrical Methods by J. L. Coolidge. The former just require high-school level but maturity. At this point you should understand elementary Euclidean and analytic geometry, properties of lines, polygons, solids, and plane conics, elementary transformations and symmetries in terms of elementary algebra. For more check out this first list.
Learn Affine and Projective Geometry: once algebrization of geometry has been introduced, see how to recover old results by the new formalism, how nicely works for higher dimensional cases and new problems, and how the complex numbers and points at infinity (motivated by perspective projections) enter the stage: Geometry by M. Audin. It is really stunning how elementary geometry can get really sophisticated: Geometry vol. I and vol. II by M. Berger. There you get the affine, projective and isometric classification of plane conic curves and [spatial] quadric surfaces over the real and complex numbers, which is the geometrical motivation to study the algebraic classification of endomorphisms and bilinear forms, among many classical topics on triangles, circles and much more. Here you will need formal linear algebra like Jänich, or like Hausner, or like Banchoff/Wermer. For more check out the same first list.
Learn Non-Euclidean Geometries and Basic Differential Geometry: check out how to generalize Euclidean geometry by dropping/changing axioms: Geometries by A. B. Sossinsky; and how non-Euclidean geometries fit nicely in curve, surface and higher-dimensional space geometries as developed by Gauß and Riemann studying their twisting and curvature to find a intrinsic, local and global, characterizations of them regardless of their ambient space: Elementary Geometry of Differentiable Curves: An Undergraduate Introduction by C.G. Gibson and Elementary Differential Geometry by C. Bär. After all this, the most sophisticated background in the major topics and concepts of differential geometry can be obtained in a very short formal course: Metric Structures in Differential Geometry by G. Walschap, where you will learn how the previous intrinsic study generalizes to abstract manifolds leading to Riemannian geometry, Kähler geometry (generalizing to the complex case), symplectic topology (generalizing classical physics phase space) and fiber/vector bundles (whose formalism is the natural framework for modern physics). For this you will also need multivariable calculus and vector analysis: Advanced Calculus: A Geometric View by J.J. Callahan. For more check out this second list.
Learn Basic Algebraic Geometry: the most important thing at first is to get what is all about, i.e. how loci defined by solutions to systems of polynomial equations generalize to abstract algebraic varieties and their regular and rational morphisms: An Invitation to Algebraic Geometry by K. E. Smith et al. and Undergraduate Algebraic Geometry by M. Reid; along with the basic abstract algebra background: A Concrete Introduction to Higher Algebra by L.N.Childs and Undergraduate Commutative Algebra by M. Reid. You may get first a good introduction by starting with just algebraic curves: Elementary Geometry of Algebraic Curves: An Undergraduate Introduction by C.G. Gibson and Complex Algebraic Curves by F.C. Kirwan. For the latter, and anyway, you will need an understanding of complex numbers and complex analysis, heuristically explained and pictured by Needham's book. For more check out this fourth list.
Learn "Classical" Algebraic Geometry: where by classical I mean anything without schemes. One of the best new, detailed and long introductions with classical constructions is: Lectures on Curves, Surfaces and Projective Varieties by Beltrametti et al., which was a fundamental motivational reference for me to grasp from where many modern concepts and constructions come from. After or alongside this, a nice algebraic companion is Algebraic Geometry: An Introduction by D. Perrin, or a more complex-analytic thorough introduction Algebraic Geometry over the Complex Numbers by D. Arapura. Great illustrative examples can be found in Algebraic Geometry: A First Course by J. Harris. From this point on, one begins to need a solid background in abstract algebra to the point of understanding cohomology, as a first exposure to sheaves and their cohomology is unavoidable, which can be studied from the beginning in a categorical style (good for getting used to more advanced algebraic geometry) using the big book Algebra: Chapter 0 by P. Aluffi or Advanced Modern Algebra by J.J. Rotman. For more check out this fifth list.
Learn Abstract Algebraic Geometry: once the concepts and formalism of algebraic varieties are understood, the natural step is towards the general formalism of schemes, where one tries to consider any commutative ring as the ring of functions on a space, and see how much geometry can still be done. I cannot help but recommend again the new book A Royal Road to Algebraic Geometry by A. Holme slowly starting from plane curves, varieties up to category theory, schemes and cohomology. The following recent books also carefully build up the theory from well-motivated examples: Algebraic Geometry 1: From Algebraic Varieties to Schemes, Algebraic Geometry 2: Sheaves and Cohomology, Algebraic Geometry 3: Further Study of Schemes by K. Ueno. The most important thing working in the abstract is understanding in detail a fair bunch of concrete examples, as developed in: The Geometry of Schemes by D. Eisenbud. For more check out this sixth list.
The final frontier: ... at this point one has mastered the background to start understanding almost any specific topic or result in modern algebraic geometry. Nice routes are: for enumerative problems, intersection of varieties and Riemann-Roch theorems go for the wonderful text: 3264 and All That by Eisenbud, Harris, and the difficult final reference Intersection Theory by W. Fulton; for arithmetic geometry (elliptic curves, abelian varieties) then maybe start with Diophantine Geometry: An Introduction by Hindry/Silverman; to start on moduli spaces: An Introduction to Invariants and Moduli by S. Mukai, and the wonderful Moduli of Curves by J. Harris; for resolution of singularities and Hironaka's theorem: Lectures on Resolution of Singularities by J. Kollár; and many other things like those at the second half of the sixth list; for birational geometry and Mori's program check out this seventh list.
This is a very long project and much material to digest, but learning algebraic geometry for real requires hard work from the very beginning and patience. The good thing is, if you are able to survive the challenge, you will end up knowing and understanding more geometry than most standard students. I have followed a similar path for self-learning algebraic geometry as a side interest for many years and worked nicely, to the point that my career goals changed.
Attempt at the revised question: the kind of problems that serve as a mathematical basis for studying the most advanced algebraic geometry are the type of questions one would like to answer for more and more sophisticated/abstract objects, which in turn tend to appear as new and deeper structures after solving and generalizing older problems. In algebraic geometry, and mathematics in general, one of the most important problems is the classification of all objects up to isomorphism or up to more manageable types of morphism, for example the birational classification of all algebraic varieties (Mori's minimal model program). This is a major research topic but can be traced back to XIXth century mathematics and even Greek geometry: given the solution points of systems of polynomial equations, when are two such geometric loci actually the same but seen upon a polynomial or rational invertible transformation? The study of all such isomorphic or birational classes of algebraic varieties, and how they behave, leads you to study moduli spaces and their geometry, which motivates the subjects of deformation theory and geometric invariant theory. As many of these things require keeping track of algebraic properties not considered classically, one is led towards schemes, algebraic spaces and stacks. You may want to know how many curves or higher-dim objects fit into other objects (enumerative geometry in Grassmannians, Schubert varieties...), or how they intersect (degree theory in Bézout's theorem...), or the properties of generalized vector fields living on them (which can be seen as physics over algebraic manifolds), all of which leads you to study vector bundles, sheaves and their number of independent sections (the Mittag-Leffler problem and Riemann-Roch theorem), characteristic classes, multiplicity in commutative algebra, homology and cohomology of cycles, differential forms and K-theory (e.g. for Grothendieck-Riemann-Roch theorem). The different types of cohomology and the structure of algebraic cycles will lead you to motives and motivic cohomology (thus to the Hodge and Grothendieck conjectures), which in turn are mysteriously related to noncommutative arithmetic geometry and the Feynman path integral from theoretical physics (where Kontsevich's motivic integration may pave the way to give a rigorous foundation of Feynman's for quantum field theories). Working with systems of polynomial equations over finite numeric fields will lead you to the problem of counting their finite number of solutions (thus to Hasse's theorem and the Weil conjectures, hence to étale topology), special types of solutions (thus to Faltings' theorem and Mordell-Weil theorem) or the generalization of Pythagoras' theorem (thus to Fermat's last theorem and the modularity theorem).
The amount of problems is immense, as all those major problems actually split up into many details and technical problems which out of context and motivation might appear discouraging to study on their own. Maybe just trying to understand what all those problems and sophisticated results state is enough motivation for learning the requisites of arid algebra. One has to realize that the most sophisticated and abstract results in algebraic geometry (or elsewhere) find their motivation in the very advanced structures where history has led mathematicians (and physicists!) to, after trying to solve easier problems. So there should be no surprise to the fact of all modern theories being highly difficult to motivate, as this happens in every old discipline, in the same sense that for a person with a very modest background in physics the problems and technical results in quantum field theory or quantum gravity are as much incomprehensible or unmotivated.
However, in my humble opinion, I think that learning a big subject just motivated by concrete [current] highly advanced problems/results is a bit off the point: the most sophisticated and abstract algebraic geometry of several centuries ago had problems and results as important to the mathematicians of the time as the contemporary ones to those of today; so why would it be a better motivation for study to understand the state of the art than the classic constructions? That is relative to one's state of knowledge, thus maybe it is more fruitful and satisfactory to try to motivate the examples, results and theory just in front of you, at your level of understanding as you gradually increase your knowledge. Eventually everything will start to fit together as you climb up to the top, and you will start to see [a small portion of] the whole mathematical landscape. There will always be places covered in fog and always a new top to conquer, that is the fun!
This doesn't answer your question, but: why don't you just start reading papers in algebraic geometry? You will quickly be forced to come to grips with "reality" in this way. My basic point is that, if you have read (a lot of) Hartshorne and Liu, you don't need more textbooks; you just need to start reading some research mathematics.
But regarding your actual question, you already have the answer at hand, namely: Hartshorne! Chapters IV and V are entirely about curves and surfaces, and have lots of concrete discussion of both of them. If you succeed in mastering this material, you will know a lot of concrete algebraic geometry.
A typical problem that it is hard to think about if you don't know anything is: "how do I describe a typical curve of genus $3$, or $4$"? After you read Hartshorne Chapter IV, you will know that the answers are "a smooth plane quartic", and "the intersection of a quadratic and cubic hypersurface in $\mathbb P^3$", respectively. You can't get much more concrete than that. (And there are many exercises in the spirit of such concrete questions.)
One thing is that you will need cohomology of coherent sheaves, but that
it not so hard to learn, and the beautiful applications in Chapters IV and V
should give ample motivation.
Best Answer
It seems that Hartshorne gets the definition of immersion "wrong." On page 120, section II.5, he defines an immersion to be an open immersion followed by a closed immersion. One might argue that a better definition is that an immersion is a closed immersion followed by an open immersion.
Hartshorne's definition creates problems, because according to his definition, a composition of immersions may not be an immersion. In particular, this makes exercise II.5.12 very awkward. See this answer for further discussion.