I believe there have been similar questions, but not one exactly of this flavor.
To answer your last question, it is true that you need to know many different areas of mathematics in order to delve deeply into algebraic geometry. On the other hand, to get a basic grounding in the field, one need only have a basic understanding of abstract algebra.
That being said, I will give my recommendations.
If you have already done complex variables, and I'm not sure that every student in your position will have completed this, I recommend Algebraic Curves and Riemann Surfaces by Rick Miranda. Although this book also develops a complex analytic point of view, it also develops the basics of the theory of algebraic curves, as well as eventually reaching the theory of sheaf cohomology. Multiple graduate students have informed me that this book helped them greatly when reading Hartshorne later on.
If you want a very elementary book, you should go with Miles Reid's Undergraduate Algebraic Geometry. This book, as its title indicates, has very few prerequisites and develops the necessary commutative algebra as it goes along. More advanced students may complain that this book does not get very far, but I think it may very well satisfy what you are looking for.
Another book you might want to check out is the book Algebraic Curves by William Fulton, which you can thankfully find online for free.
If you would not mind a computational approach, and furthermore a book which requires even fewer algebraic prerequisites than you seem to have, you might want to check out Ideals, Varieties, and Algorithms by Cox and O'Shea.
Thierry Zell's suggestion is also supposed to be good.
That being said, if you decide that you like algebraic geometry and decide to go more deeply into the subject, I highly recommend that you learn some commutative algebra (such as through Commutative Algebra by Atiyah and Macdonald). But for the moment, I think the above recommendations will suit you well.
I propose the following plan, assuming a basic background in scheme theory and algebraic topology. I assume that you are interested in derived algebraic geometry from the point of view of applications in algebraic geometry. (If you are interested in applications to topology, you should replace part 2) of the plan by Lurie's Higher algebra.) The plan is based on what worked best for myself, and it's certainly possible that you may prefer to jump into Higher Topos Theory as Yonatan suggested.
0) First of all, make sure you have a solid grounding in basic category theory. For this, read the first two chapters of the excellent lecture notes of Schapira. I would strongly recommend reading chapters 3 and 4 as well, but these can be skipped for now.
Then read chapters I and II of Gabriel-Zisman, Calculus of fractions and homotopy theory, to learn about the theory of localization of categories.
1) The next step is to learn the basics of abstract homotopy theory.
I recommend working through Cisinski's notes. This will take you through simplicial sets, model categories, a beautiful construction of the Quillen and Joyal model structures (which present $\infty$-groupoids and $\infty$-categories, respectively), and the fundamental constructions of $\infty$-category theory (functor categories, homotopy (co)limits, fibred categories, prestacks, etc.).
Supplement the section "Catégories de modèles" with chapter I of Quillen's lecture notes Homotopical algebra.
Then read about stable $\infty$-categories and symmetric monoidal $\infty$-categories in these notes from a mini-course by Cisinski. (By the way, these ones are in English and also summarize very briefly some of the material from the longer course notes). These notes are very brief, so you will have to supplement them with the notes of Joyal. It may also be helpful to have a look at the first chapter of Lurie's Higher algebra and the notes of Moritz Groth.
2) At this point you are ready to learn some derived commutative algebra:
Read lecture 4 of part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry together with section 3 of Lurie's thesis. Supplement this with section 2.2.2 of Toen-Vezzosi's HAG II, referring to chapter 1.2 when necessary. This material is at the heart of derived algebraic geometry: the cotangent complex, infinitesimal extensions, Postnikov towers of simplicial commutative rings, etc.
Other helpful things to look at are Schwede's Diplomarbeit and Quillen's Homology of commutative rings.
3) Before learning about derived stacks, I would strongly recommend working through these notes of Toen about classical algebraic stacks, from a homotopy theoretic perspective. There are also these notes of Preygel. This will make it a lot easier to understand what comes next.
Then, read Lurie's On $\infty$-topoi. It will be helpful to consult sections 15-20 of Cisinski's Bourbaki talk, section 40 of Joyal's notes on quasi-categories, and Rezk's notes. For a summary of this material, see lecture 2 of Moerdijk-Toen.
4) Finally, read about derived stacks in lecture 5 of Moerdijk-Toen and section 5 of Lurie's thesis. Again, chapters 1.3, 1.4, and 2.2 of HAG II will be very helpful references. See also Gaitsgory's notes (he works with commutative connective dg-algebras instead of simplicial commutative rings, but this makes little difference). His notes on quasi-coherent sheaves in DAG are also very good.
5) At this point, you know the definitions of objects in derived algebraic geometry. To get some experience working with them, I would recommend reading some of the following papers:
- Antieau-Gepner, Brauer groups and étale cohomology in derived algebraic geometry, arXiv:1210.0290
- Bhatt, p-adic derived de Rham cohomology, arXiv:1204.6560.
- Bhatt-Scholze, Projectivity of the Witt vector affine Grassmannian, arXiv:1507.06490.
- Gaitsgory-Rozenblyum, A study in derived algebraic geometry, link
- Kerz-Strunk-Tamme, Algebraic K-theory and descent for blow-ups, arXiv:1611.08466.
- Toen, Derived Azumaya algebras and generators for twisted derived categories, arXiv:1002.2599.
- Toen, Proper lci morphisms preserve perfect complexes, arXiv:1210.2827.
- Toen-Vaquie, Moduli of objects in dg-categories, arXiv:math/0503269.
Best Answer
This is not really an answer to your question, just an attempt to address your question from the comments.
There are various flavours of homotopical or higher algebraic geometry that are commonly considered, which have different levels of connectivity, linearity, or strictness of commutativity. These include:
1) simplicial commutative rings
2) connective $E_\infty$-algebras over $H\mathbf{Z}$
3) connective $E_\infty$-ring spectra
4) (nonconnective) $E_\infty$-algebras over $H\mathbf{Z}$
5) (nonconnective) $E_\infty$-ring spectra
In characteristic zero, one also considers (connective) commutative dg-algebras.
The flavour most suited for algebraic geometry purposes is (1): this is the minimal extension of algebraic geometry where derived tensor products and cotangent complexes live. This was the flavour originally studied by Lurie in his thesis, and Toen-Vezzosi in HAG II.
Any of the other theories might be called "spectral algebraic geometry". (2) is similar to (1), but is less suited for algebraic geometry purposes, because deformation theory in the $E_\infty$-world is different than in the setting of simplicial commutative rings. In fact, the affine line is not even smooth in the $E_\infty$-world.
The difference between (2) and (3), as between (4) and (5), is linearity: in (3) and (5), you only consider objects which are linear over the sphere spectrum, so these settings are well suited to purposes of stable homotopy theory.
The main difference between the connective and nonconnective settings is the lack of converging Postnikov towers. That is, every connective $E_\infty$-ring spectrum $R$ can be written as a homotopy limit of square zero extensions of $\pi_0(R)$. This allows one to establish analogues of many results from classical algebraic geometry, by using induction along square zero extensions. The nonconnective world, on the other hand, behaves much differently, and geometric intuition very often fails.
I don't know much stable homotopy theory, but I believe the main point of spectral algebraic geometry is to be able to consider $E_\infty$-ring spectra as affine schemes, and to apply algebro-geometric techniques to study them. For example, the main application so far is Lurie's construction of tmf, the spectrum of topological modular forms, as the global sections of a sheaf of $E_\infty$-ring spectra on the moduli stack of spectral elliptic curves.