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.

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). 


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... 

