I don't know much about this stuff, so instead of answering the question, I try to formulate more precise questions, in the hope someone else will take up these questions:
One of the main reason to look for cycles is that they give realizations (their fundamental class) in all cohomology theories, which happen to have special properties (e.g., are Hodge cycles or Tate cycles), and anytime you see a Hodge (or Tate) cycle in cohomology, you expect that it comes from an algebraic cycle (the Hodge or Tate conjecture) and hence similar phenomena should occur in all cohomology theories (i.e., there is a Hodge (or Tate) cycle in all realizations).
Now, if the following were true:
1) Any 'derived algebraic cycle' gives rise to virtual fundamental classes in all cohomology theories, which again are Hodge or Tate cycles.
2) It is not clear that the virtual fundamental classes of 'derived algebraic cycles' are already fundamental classes of real algebraic cycles,
then one might formulate a 'derived' Hodge or Tate conjecture, which would have the same consequences.
Your question has another aspect, which regards a possible framework for working with these motives; I leave this aside as I understand even less about how this should work.
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.
Best Answer
I would add this as a comment, but I do not have enough reputation to do so.
While there are certainly more contemporary references, Voevodsky's "Triangulated category of motives over a field" is a place where you can read about motives (https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/s5.pdf). His paper with Morel on $\mathbb{A}^{1}$-homotopy theory would be a source to address your interest in homotopy theory (http://www.math.ias.edu/vladimir/files/A1_homotopy_with_Morel_published.pdf). A classical reference is Manin's article "Correspondences, motifs, and monoidal transformations."