Where can I find good exercises for algebraic geometry that require hard, concrete computation? 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?

 A: This doesn't answer your question, but: why don't you just start reading papers in algebraic geometry?  You will quickly be forced to come to grips with "reality" in this way.  My basic point is that, if you have read (a lot of) Hartshorne and Liu, you don't need more textbooks; you just need to start reading some research mathematics.

But regarding your actual question, you already have the answer at hand, namely: Hartshorne!  Chapters IV and V are entirely about curves and surfaces, and have lots of concrete discussion of both of them.  If you succeed in mastering this material, you will know a lot of concrete algebraic geometry.
A typical problem that it is hard to think about if you don't know anything is: "how do I describe a typical curve of genus $3$, or $4$"?  After you read Hartshorne Chapter IV, you will know that the answers are "a smooth plane quartic", and "the intersection of a quadratic and cubic hypersurface in $\mathbb P^3$", respectively.  You can't get much more concrete than that. (And there are many exercises in the spirit of such concrete questions.)  
One thing is that you will need cohomology of coherent sheaves, but that 
it not so hard to learn, and the beautiful applications in Chapters IV and V
should give ample motivation.
A: If you feel like you have a good command of the material you've seen in Hartshorne but now you want to use it more specifically, Matt has answered your question perfectly. Reading papers and the latter chapters of Hartshorne will give many applications of the theory you've seen developed. If however your command of the material is not as good as you'd like and you want to look examples and calculations to improve your intuition, then I've listed some sources that I think can help. In fact, even if you aren't struggling, these books have many interesting concrete exercises.
Three texts that present their material in a very concrete fashion:


*

*Algebraic Curves by Fulton

*The Geometry of Schemes by Eisenbud and Harris

*Schemes with Examples and Exercises by Görtz and Wedhorn.


I am not certain of what type of problems you want so I included Fulton which is an elementary introduction to classical algebraic geometry (no schemes). Despite it developing the theory through elementary methods, you can still find problems involving concrete curves that I think you will find interesting. Look especially in Chapters 3 (Multiplicities, intersection numbers), 5 (Bezout's theorem, Max Noether's theorem) and 8 (Riemann-Roch). 
The other two are at about the same level as each other, which is a presentation of the basics of schemes in a more concrete form than Hartshorne. Lots of examples and exercises that will build intuition in both. Neither cover cohomology however, but Matt says Hartshorne gives nice applications of cohomology in the latter chapters so you should be fine that for. 
A: you can consult the book "Computational Commutative Algebra and Algebraic Geometry" https://www.amazon.fr/Computational-Commutative-Algebra-Algebraic-Geometry-ebook/dp/B07RB6DDVC
It contains It contains a total of 124 exercises (15 on Gröbner bases over arithmetical rings, 11 on Varieties, Ideals and Gröbner bases, 19 on Finite fields, 11 on Algorithms for cryptography, 33 on Algebraic plane curves, and finally 35 on Elliptic curves) with detailed solutions.
