I have a 6-yr-old and a 3-yr-old. So far the smaller one is happy enough with a few kid-friendly games on mom's iPhone (very sparingly, a few times a week at most) but I'm finding the 6-yr-old very engaged with online casual puzzle games. He's not quite ready for escape-the-room-type stuff, but there are quite a few kid-friendly puzzlers out there - check JayIsGames.com, you can search by tags and I use "kidfriendly" and "puzzle" (and "flash" because I'm not downloading anything, even if I did have Windows around)
While others may disagree, I'm happy to let the local schools teach the basic 3 R's; later on I'll supplement the history and geography and science. Right now I'm more interested in making sure he has analytical skills, including skeptical thinking and inductive reasoning. He's not the math/science geek I was at his age. So I'm trying to make sure he learns as much as possible about puzzles and different ways of solving them. "Rubble Trouble" is a current favorite, but that's only after we've gone through most of the Bonte stable (especially "Factory Balls"! he solved virtually all of them, not bad for (at the time) a 5-year-old)
As for the 3-yr-old, he loves Angry Birds first and foremost, but is just as happy tossing the birds to the left as to the right. He's developing differently and takes things at his own speed, so we're kind of feeling our way forward. (by contrast with the older boy, who's basically a carbon copy of my wife's personality) If he ends up more like me there'll be no keeping him away from the more analytical, strategy-type games - violence-based or otherwise - and he'll be wanting to modify the games as soon as he's finished with 'em... I'll certainly be showing him puzzles of every shape and size until we find a genre he likes.
Kids can only be steered so far. But there's enough out there that you can find something that you and the child can agree on for almost any combination of "you" and "the child"!