FGA Explained. Articles by a bunch of people, most of them free online. You have Vistoli explaining what a Stack is, with Descent Theory, Nitsure constructing the Hilbert and Quot schemes, with interesting special cases examined by Fantechi and Goettsche, Illusie doing formal geometry and Kleiman talking about the Picard scheme.
For intersection theory, I second Fulton's book.
And for more on the Hilbert scheme (and Chow varieties, for that matter) I rather like the first chapter of Kollar's "Rational Curves on Algebraic Varieties", though he references a couple of theorems in Mumfords "Curves on Surfaces" to do the construction.
And on the "algebraic geometry sucks" part, I never hit it, but then I've been just grabbing things piecemeal for awhile and not worrying too much about getting a proper, thorough grounding in any bit of technical stuff until I really need it, and when I do anything, I always just fall back to focus on varieties over C to make sure I know what's going on.
EDIT: Forgot to mention, Gelfand, Kapranov, Zelevinsky "Discriminants, resultants and multidimensional determinants" covers a lot of ground, fairly concretely, including Chow varieties and some toric stuff, if I recall right (don't have it in front of me)
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.
Best Answer
Wrong!
Here is Bourbaki document on algebraic geometry, taken from the now available Master's Archives: click on Autres rédactions, then on Chap.I Théorie globale élémentaire (91 p.)
This preliminary draft was apparently written (according to a penciled annotation on the first page) for Bourbaki in 1954 by Samuel, a distinguished algebraic geometer and number theorist.
Alas, it is hard to conceive a worse timing for a book on algebraic geometry: one year later Serre would publish his paradigm shifting FAC, shortly followed by Grothendieck's theory of schemes, a vast development of Serre's article (as acknowledged in the Preface to the EGA), which would forever change our vision of algebraic geometry.
Samuel's point of view is that of Weil: at the forefront is a "universal domain", a field extension $k\subset K$ with $K$ algebraically closed and of infinite transcendency degree over $k$.
Geometry would happen in $\mathbb A^n(K)$ or $\mathbb P^n(K)$, whereas algebra and number theory would take place inside $k[T_1,\dots,T_n]$ or $k[T_0,\dots,T_n]$.
A variety in Weil's vision could have a multitude of generic points, essentially points such that a polynomial vanishing on them must be zero.
It is quite moving to see the author struggling with, for example, the product of varieties: he notices the difficulty due to the tensor product $E\otimes_k F$ of two field extensions $k\subset E,F$ having non-zero nilpotents but doesn't envision incorporating these in his foundational text.
Grothendieck would soon show the world how considering nilpotents in the very foundations of scheme theory would enrich and beautify algebraic geometry.
I encourage every algebraic geometer to browse this nostalgic and unacknowledged witness of a bygone era of our beloved science.
Edit (May 27th, 2016)
Browsing the fascinating Grothendieck-Serre Correspondence ( a review of which is here) I found this excerpt from the very first letter of the Correspondence (page 3, dated January 28th, 1955), written by Grothendieck then in Lawrence, Kansas, USA :
This confirms that the document mentioned above was indeed authored by Samuel.