What I don't like about these games is that the outcome is largely random - in essence, the games almost play themselves. I'm wondering if there are good examples of early games that aren't essentially random. I might try out backgammon and checkers, but those are quite a step up from dominoes. I think backgammon would be easier to teach, since its long-term goal is easier to understand and translate into short term goals.
An interesting question is whether chess is right out at this point. Is it good to introduce the pieces at an early age so that children can internalize their movements, or to hold off to the point where they might be able to grasp some of the subtleties of the game. I remember being taught chess as a child and thinking that since I knew how the pieces moved, I knew how to play. Nothing, obviously, could be further from the truth.
Now that it's summer there will likely be a lot of time for game playing. What games did you like to play as a child? Which of them were random and which taught that choices affect outcomes?