Just starting with [combinatorial] game theory I have recently become interested in game theory by way of John Conway's on Numbers and Games.  Having virtually no prior knowledge of game theory, what is the best place to start?
 A: Extra sources:
"Lessons In Play" fills in most of the mathematical details for volume one of "Winning Ways"; 
"Games of No Chance 3" is out and is available online through MSRI publications;
Aaron Siegel's upcoming book on combinatorial games will include all the advances in the theory, he's responsible for many, and will be a good companion to "On Numbers and Games"
A: First, "game theory" (as used in economics) and "combinatorial game theory" are two completely different things.
Second, I think the best place to start learning combinatorial game theory is still Winning Ways for your Mathematical Plays by Berlekamp, Conway, and Guy.
A: In addition to Winning Ways for your Mathematical Plays, I also recommend the books Games of No Chance and More Games of No Chance. Many of the articles from Games of No Chance are online.
A: If you have played Go, then you may enjoy Mathematical Go: Chilling Gets the Last Point by Berlekamp and Wolfe. Some go players, especially Bill Spight, have developed combinatorial game theory within go further. 
A: Winning Ways for your Mathematical Plays (in four volumes) has an enormous amount of stuff about combinatorial games. But most of it you probably won't be interested in for a while. There are a few quickly diverging directions one could study in combinatorial games. Here are some that come to mind immediately, and a possible list of topics to study in each:
1) Impartial games. Read a bit of On Numbers and Games so that you know how to read the notation and understand game equivalence and addition. Then learn the winning strategy for Nim, read the relevant bits of Chapter 3 and all of Chapter 4 of Winning Ways. After that, if you like the infinite theory, Lenstra has a paper called "On the algebraic closure of two" which is really nice. If you like the finite theory, learn about nim multiplication from ONAG, and then read Conway and Sloane's paper "Lexicographic codes: error-correcting codes from game theory." I think this part of the theory is the most interesting.
2) (Surreal) numbers. Again, learn how to read the notation and about game equivalence and addition. (You will need this for everything.) Then read the first part of ONAG. Then, perhaps learn about real-closed fields in general; you can make most of real analysis work over the Field of surreal numbers. (A Field is something like a field, but it has a proper class of objects instead of a set.)
3) Weird games, for example from Hackenbush and Domineering. Read Volume 1 of Winning Ways. The stuff on thermography and all-small games is quite interesting.
