Good way to learn Ramsey Theory
This is a list of some resource material on Ramsey Theory:
Expository note on Arithmetic Ramsey theory by Terence Tao. This contains the basic theorems and proofs presented in a very nice manner.
Ramsey Theory by R.L. Graham, B.L. Rothschild, and J. Spencer is a book which contains an introduction to nearly all areas in finite Ramsey theory. It contains proofs of all the basic results: Ramsey's theorem, van der Waerden theorem, Hales Jewett theorem, Schur's theorem, Rado's theorem, Graham's theorem etc.
Ramsey theory on the integers by Landman is a book accessible to undergraduates. It is unique in the sense that it uses only elementary mathematics and describes many research problems. Most of the book deals with variants of the van der Waerden's theorem.
Mathematics of Ramsey theory by Nesteril & Rodl is a book which I haven't read but is supposed to contain many advanced techniques.
The ebook Introduction to Graph Ramsey Theory available here.
Elemental methods in Ergodic Ramsey theory by McCutcheon introduces one of the current approaches to Ramsey theory (via Ergodic theory). I have only read the first chapter but it seems an accessible and well written book.
Erdos' book Partition Relations for Cardinals (Infinite Ramsey theory). I haven't read it.
I still like the first edition of R.L. Graham, B.L. Rothschild, and J. Spencer, Ramsey Theory, from 1980. There was a second edition in 1990 that I’ve not seen but which I believe added quite a bit of new material. Unfortunately, it’s obscenely expensive.
I have not seen B.M. Landman and A. Robertson, Ramsey Theory on the Integers, but it’s received some very good reviews and sports a much saner price tag.
You might like Ronald L. Graham's Rudiments of Ramsey Theory, which is quite short, and which I think is intended to be an introduction. I haven't looked at it myself in many years, so I can't provide a more detailed review than that.