I am trying to learn formal schemes. I tried to read the section in Hartshorne but I don't get very far from there since things are not done quite explicitly enough, at least in my opinion. I cannot read French, so EGA is out of the question. I would really appreciate it if you could tell me a good introduction to this topic.

2$\begingroup$ A free draft version of the Illusie chapter BCnrd suggested is available at (cdsagenda5.ictp.it//…). $\endgroup$ – Evan Jenkins Dec 2 '10 at 4:04

16$\begingroup$ Read Illusie's exposition (in English!) of the important and aweinspiring results from EGA III$_1$ in the book "FGA Explained". That should clear up everything (motivation, technique, ways to think about and work with them, etc.). Note that the deepest results are in the proper case (just like GAGA). By the way, practice the math French; well worth the effort, and needs virtually no knowledge of real French. $\endgroup$ – BCnrd Dec 2 '10 at 4:04

2$\begingroup$ Perhaps you may try EGA I new edition after all. You will have to struggle with the language for a week, but after that you'll realize that the sentences are of a few types and the vocabulary is relatively limited. The exposition there is complete and lucid, so the effort will pay off. $\endgroup$ – Leo Alonso Dec 2 '10 at 9:56

1$\begingroup$ this may seem nuts, but you probably can read french if you try. e.g. "theoreme" = theorem. "epreuve" = proof = "demonstration", "algebrique" = algebraic,.... you do need a few verbs, but it is well worth the effort to learn them. take my word for it, give it a few more minutes... e.g. start with serre's fac. $\endgroup$ – roy smith Dec 3 '10 at 5:44

1$\begingroup$ I once tried looking up "conoyau" before seeing "noyau" (= kernel). That was fun. $\endgroup$ – Harry Gindi Dec 5 '10 at 17:26
I really liked this paper by Neil Strickland. It's quite good, but it only really covers the affine case. Be forewarned that, as stated in the introduction, scheme is taken to mean affine scheme unless explicitly noted. A very nice thing about the approach of this paper is that it works without many/any finiteness assumptions (since these often cannot be assumed for algebrotopological or homotopytheoretic applications).

7$\begingroup$ (Since Neil visits MO, I'm going to use this comment to request that he write a sequel that develops the theory of nonaffine schemes or algebraic spaces using the same functorial framework used in this paper). $\endgroup$ – Harry Gindi Dec 2 '10 at 3:53
If you are interested in the infinitesimal structure of formal schemes (with emphasis on non necessarily adic maps) I suggest to look at my papers with Jeremías and Pérez "Infinitesimal Lifting and Jacobi Criterion for Smoothness on Formal Schemes" (Comm. Alg.) and "Local structure theorems for smooth maps of formal schemes" (JPAA). Also for the deformation theory of formal schemes look at M. Pérez "Basic deformation theory of smooth formal schemes" J. Pure Appl. Algebra 212 (2008), pp. 2381–2388 (MR 2009h:14006). I apologize for this selfpromotion...