I've been studying scheme theory from Hartshorne and Qing Liu for a few months now. (For those who are not big fans of Hartshorne, I have to note that I agree with you: I use it only for exercises.) I now have a basic understanding of concepts like separatedness and properness, and quasicoherent sheaves, although I haven't gotten to cohomology quite yet.
I do all the exercises that don't seem totally trivial (and there are indeed a few of those, in both textbooks). I do feel like this has given me a good solid understanding of the material, but I also note that it's a bit one-sided. All the exercises in both texts are pretty abstract. I do what I think makes the most of them: for each exercise, I tend to write 5-10 page solutions which carefully develop machinery to makes the problem trivial, and this has been great practice for the broad realities of research. Typically a term will come to mind that I've heard but never studied and which seems applicable, and I'll develop that concept (or what I think that concept ought to be) until it solves my problem.
At risk of repeating it too much, I'm really happy with certain aspects of what I've gotten out of this, but it's also clearly a very impoverished approach: one unfortunate result is that I rarely get to sit down with a real scheme and do some real computation and explore a real example. I'm becoming increasingly aware of the fact that if I were faced with such a situation, I wouldn't know where to begin.
So here's my question.
Hartshorne and Qing Liu have essentially the same issue. I've also used Vakil's FOAG, which is markedly better, but still a bit weak in that department. Where can I find a collection of exercises in algebraic geometry and schemes that will force me to get my hands dirty with some real schemes, and really compute something?