References to learn $f(R)$ gravity? What reference(s) can I use to learn $f(R)$ gravity?
 A: The most comprehensive text, in my experience, is "Extensions of f(R) gravity" by Harko and Lobo. It's part of the Cambridge Monographs on Mathematical physics, so you know it's of a high quality. It runs through some of the basics of GR and constraints on modified gravity theories (eg. Solar system tests etc.). It has a fairly lengthy, good chapter on basic f(R) gravity and progresses onto more complicated extended theories and applications in cosmology etc. 
I would recommend a good working knowledge of GR and variational identities within GR, it's also a pretty good idea to be familiar with QFT in curved space and the Palatini formalism. The book does go over these topics but topics like QFT in curved space are so pivotal to theoretical physics you should know them before you look at f(R) gravity, especially if you're interested in cosmological models or black holes etc. 
Also, there are a wealth of introductory papers on the topic. This paper, gives a great intro to f(R) and scalar-tensor theories (they are very closely related).
