I am a game developer, and probably guilty of over-exposing my two daughters to video games. I make an attempt to limit the amount of time they get with digital devices, but introduced them both to touch devices at a young age. (At least we don't have a TV in the house! I'm more against exposure to mass media advertising than screens per-se.) My older daughter learned to use a d-pad style controller using an ipad (with a simple platformer), and was able to learn to use a playstation controller with a little practice, once her hands were big enough to hold it.
I think the most important thing with kids and games is to be part of the play as much as possible. Don't just leave them there to play alone. Play with them, teach them, help them. This can be hard with touch devices, which tend to isolate the player in their own world, which is why I recommend playing co-op games on a console.
Racing games are fun, but with my own kids I've noticed that they are actually somewhat harder than platform games. There is a lot of control involved, and when you fall behind, or crash, there is a high frustration level. This is why I encourage co-op play where going slow and helping your child is a defined goal of the game.
In my opinion, the best and most fun game to play with kids is Little Big Planet 2. The game is visually fun and quite forgiving about controls. It is easy to find simple user-created levels online that don't require any technical platforming, with a focus on exploration, problem solving, or roleplaying. Because the game is co-op, you can guide and assist your child by grabbing onto them in-game, or having them hold on to you while you swing / grapple / swim etc. There is a creative aspect to the game as well. You can dismantle the levels and move items around. Sometimes my daughter will spend ages just dressing up, or decorating the "pod". For older kids, there is a creative mode where you can create entire levels. This is a lot of fun, and you could also make special customised levels for your own child if you want to.
A note on game genres: Personally, I have a strong preference for letting my children play fast paced / coordination based games than "casual" games. While they may often seem mentally stimulating, they are frequently engineered to produce addiction using psychological techniques originally developed for gambling machines. The mobile market is saturated with these kinds of games. Be especially wary of free-to-play casual games. I tend to think it is better to pay for a game with a pay-up-front business model, than one one that comes free but uses psychological manipulation to try to get money out of people. (Especially children.)