My kids haven't quite gotten the hang of board games yet: They find them enticing, but the cherries from "Hi Ho Cherry-O" were quickly scattered throughout our apartment; the "Candyland" board got ripped in half.
The lesson, for me? Simplify.
So we've been playing little games with dominoes and dice. My kids are both growing more interested in counting and pattern recognition, and these are simple, fun ways to nurture these interests. Our games are non-competitive, because my kids remain happily ignorant of the concepts of "winning" and "losing." (They'll have far too many opportunities in life to learn these things.)
The domino game, for us, is a simple matching game; with dice, they roll and count out small objects, like mini poker chips. My daughter tends to modify the activities: She quickly decided that the dominoes were "butter," and got engrossed in building butter stacks; having recently attended the Little Red Lighthouse Festival, she stopped rolling the dice after a few minutes and started building little red lighthouses with the poker chips instead.
My son, on the other hand, is very intent on the ordering and counting and organizing. Both of them are increasing their facility with numbers and counting and patterns, in a low-key, low-budget, non-pressured way.