Since you mentioned tic-tac-toe, I've worked along these same set of assumptions (know what's a legal move, know if & how the game ends) for both (very simple) neural nets and also genetic algorithms. Both a lot of fun, you may want to look into them. I must admit that the math for the NN is very slow going when it's not beyond me, but GAs were fun and cool.

Two twists on it that I tried I think may of helped, or at least kept me more interested. For games like tic-tac-toe where board orientation is unimportant I developed a routine to orient the board before evaluating such that multiple boards that were just rotations or flips of each other went through the same game logic.

In addition, I started with many randomly weighted oppenents, and then let them battle each other. This slowly gave a "better opponent" as you mention, which I think allowed a quicker training. It was needful for the GAs, but just plain fun for the NNs.

There are a lot or resources out there on the web. My old bookmark list (that had my favs) got wiped a while ago so I can't rememebr specific ones, but just search around.

Have fun.

=Blue
...you might be eaten by a grue...