I have some suggestions. However, to properly understand Game Theory, a strong background in Mathematics in not strictly necessary on the beginning, in which a basic knowledge on calculus and algebra is sufficient. Of course that as you advance, that strong background will be a difference maker.
You don't give me too much information about the knowledge you already possess on the topic. However, the best choice for standard non-cooperative game theory (static and dynamic games, in complete and incomplete information) is clearly:
- Fudenberg, D. and J. Tirole (1991). Game Theory. MIT Press
Despite what others may argue, this is the most rigorous book on the area. Then if you want to continue, the following books nicely complement the previous:
Cooperative Game Theory (coalitional and matching) (introductory/medium level):
Osborne, M. and A. Rubinstein (1994). A course in Game Theory. MIT Press (book available online and for free at http://arielrubinstein.tau.ac.il/), Ch. 13-15
Roth, A. and M. Sotomayor (1990). Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis. Cambridge University Press
Repeated Games, Equilibrium, Structure of Games (medium/advanced level):
Mertens, J-F, Sorin, S. and S. Zamir (2015). Repeated Games. Cambridge University Press.
Van Damme, E. (1991). Stability and Perfection of the Nash Equilibrium. Springer.
Epistemic Game Theory and Logic (medium/advanced level):
- Perea, A. (2012). Epistemic Game Theory. Cambridge University Press
(note: I am not highlighting some topics that I may appreciate. Just addressing those that are not cover in detail on the first book, and that, if they capture your attention, should be complemented).
This will cover more or less what is nowadays considered a robust knowledge of Game Theory. Other topics, like Algorithmic Game Theory, Behavioral Game Theory, and so on should be addressed only on a later stage.