I asked this same question about a year ago, so I'm very slightly ahead of you. Here's what I know:
As you probably know, there are two major branches for game theory. There's (for lack of a better term) "economics" game theory dealing with real world situations, economics, politics and the like. I know next to nothing about that. However, I do know a decent amount about combinatorial game theory, which is a little bit more ground in mathematics, and deals with two player games such as Go, Chess, Nim, or Tic-Tac-Toe.
The best introductory text is going to be Conway's Winning Ways, any of the volumes 1-4. These are the books to read to get into any other subset of combinatorial games, in my opinion.
My personal specialization thus far is generalizations of Tic-Tac-Toe called achievement games, which you can read about (along with much more) in Tic-Tac-Toe Theory.
However, if you want to go even further in these studies, you are a little bit out of luck. What's very exciting to me about combinatorial game theory is that it's pretty much a brand new field of mathematics, and right now the best techniques we have to study it are educated guessing/brute-forcing and a little bit of discrete mathematics. Although it's disenchanting sometimes, this also means that there is potentially a world of possible links and connections to other branches of math that we don't know about, and is just out there waiting to be discovered.