Good introductory references on moduli (stacks), for arithmetic objects
If you want to learn about stacks, I can recommend 'Fundamental Algebraic Geometry: Grothendieck's FGA Explained'. Vistoli's exposition of the basic theory of stacks is hard to beat, I think. Moreover, the chapter about Picard schemes is also good if you want to learn when a functor is representable and what you might have to do to make it representable. In any case, the theory of Picard schemes is indispensable in arithmetic geometry, e.g. it is a way to obtain the group structure on an elliptic curve (see the book of Katz, Mazur: Arithmetic moduli of elliptic curves) or to study duality of abelian varieties.
I am not sure if there exists an ideal book if you want to learn about the moduli stack of elliptic curves. The classic source is Deligne-Rapoport 'LES SCHEMAS DE MODULES DE COURBES ELLIPTIQUE', which you should look into, but the proofs are often very brief. The book by Katz and Mazur has good parts, but for some reason they decided to avoid the language of stacks. The book by Olsson on stacks has some parts about elliptic curves as well. At some point, I wrote a note together with Viktoriya Ozornova that contains a really detailed proof that the moduli stack of elliptic curves is really an fpqc stack and that Weierstraß equations exist http://www.staff.science.uu.nl/~meier007/Mell.pdf (but it is certainly not meant to provide a flair of the subject).
A very nice introduction to moduli (stack) of elliptic curves is R. Hain - Lectures on Moduli Space of Elliptic Curves (you find it on arXiv):
- it is completed by 100 and more exercises;
- a brief appendix is devoted to stacks (I suggest to integrate it with Vistoli's lecture notes on stacks);
- it doesn't introduce the elliptic curves, therefore you must know it (but this is not the case of the OP).