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)
Sturmfels/Speyer may be a good start. Otherwise I would look around on Arxiv particularly things that Dr. Bernd Sturmfels has written. Also, if you peruse his website (just look up Bernd Sturmfels + UC Berkeley), you'll probably get something interesting.
Best Answer
Several years ago, I participated in a learning seminar in tropical algebraic geometry and collected several helpful survey articles. (This was before Maclagan and Sturmfels' book was written, which I suspect is excellent.)
Anyway, here were some of the most helpful intro points for me: Tropical Mathematics, First Steps in Tropical Geometry, Tropical Algebraic Geometry, Introduction to Tropical Geometry, The Tropical Grassmannian, The Number of Tropical Plane Curves Through Points in General Position.
Sturmfels, Speyer, and Gathmann all write very well, and Gathmann especially devotes considerable space to giving motivation for the field. Mikhalkin, of course, was the one who pioneered the idea of attacking challenging classical problems (such as counting the number of plane curves of genus $g$ and degree $d$ passing through $3d + g - 1$ points, which had just been solved by Capraso-Harris in the late 90s) using the tropical semifield.