Tic-Tac-Toe is easy to solve; in this game, the result will be a tie if both players are playing optimally.
Connect Four has been solved by computer program; I believe the result is the first player always wins.
Chess has not been solved; because one player's move can make other moves by the other player possible (something not the case in Tic-Tac-Toe), it's not safe to assume that the result isn't "player 2 always wins".
A board game called "Go" is apparently much harder than chess.
Some versions of the game "Nim" are guaranteed wins for the second player.
The game where there are 21 tokens, you take 1 to 4 on your turn, and whoever takes the last one *loses* is a win for the second player. (Interestingly, Super Mario RPG puts you into this situation at one point, but the AI is (deliberately) imperfect, so once it makes a mistake, you can easily win.)
Anyway, here is a relatively simple programming exercise: Write a program that plays Tic-Tac-Toe optimally. It should never lose, regardless of what the other player does.
For actually playing the game, however, it would be more fun to play against a program that plays randomly rather than optimally.