There are many books on game theory. I am mostly familiar with those written by economists. The fact that books listed below are (mostly) written by economists does not mean they are not rigorous! But it means that certain aspects are taken for granted, for example, in the choice of examples covered.
- Osborne "An Introduction to Game Theory" - Probably the most comprehensive introduction to game theory. It is undergraduate but uses math.
- Gibbons "Game Theory for Applied Economists" - Another good introduction.
These are probably the most mathematical and most complete undergraduate textbooks (though I might be unfamiliar with some newer ones).
There are three classic textbooks for graduate level game theory.
- Fudenberg and Tirole - Game Theory
- Myerson - Game Theory: Analysis of Conflict
- Rubinstein and Osborne - A Course in Game Theory
Out of these three I personally recommend the last one. It is the shortest of the three, but the most elegant. It covers standard topics taught to PhD students in economics in the 1st year of PhD and you might be able to get it for free from Ariel Rubinstein's website. Fudenberg Tirole covers broader set of topics than Rubinstein and Osborne. Importantly, it covers mechanism design and auctions (also core topics). But is covers a lot of material and some it is is outside of what I would call the "core". I have not used the book by Myerson so can't comment on it, but heard it is a nice companion to either of the two.
More advanced books:
- Maliath Samuelson - Repeated Games and Reputations: Long-Run Relationships
- Bolton Dewatripont - Contract Theory
- Zamir, Maschler & Solan - Game Theory
- Krishna - Auction Theory
Once you learned the basic of game theory it is time to choose more specialized topics. Maliath and Samuelson focuses on a more recent developments in repeated games with particular focus on the role of reputation. Bolton and Dewatripont, as the name suggests, focus on design of optimal contracts. The book Zamir, Maschler & Solan is a great modern reference. It is really a encyclopedia of Game Theory and I would never suggest to anyone going through all of it. I doubt thee are more than a handful researchers that know ALL of that stuff. But it is a great way to get a quick start on a particular topic of interest. Finally, the book by Krishna is the reference for Auction theory, which has found a lot of applications outside academia.
Another good reference is the "Handbook of Game Theory" which consists of three volumes. Again, it is really a reference for researchers who want a quick introduction to a particular topic rather than a textbook to learn from it.
P.S. This guide is written from a perspective of economist.