He has played Uno (without the scoring and with face up hands of cards) for a while now. The other game he has been able to play and that I recommend as not being as painful for adults as some, is an Orchard Toys game called 'Where's my cupcake?', but he's only just at the snakes and ladders type stage. We played a lot of Sorry too with my six year old when he was about 4-4.5. The other thing we have done is having the three year old 'on a team' with an adult, but that obviously works better with multiple kids.
One they hit 4.5/5, there are tons of games you can play with them. That's when things start get to fun! Basically, I wouldn't worry yet - they just have to get to the right stage developmentally which they will do.
At first, early on, games should be very simple. The idea is to focus solely on the basic structure of games:
--game start
--keep attention on game
--turn taking/following in correct sequence
--problem solving around a very simple goal
--game end
--with finally very clear assessment of player ranking (i.e. You won, I/we lost or I won, You lost)
Underlying all this should be a heavy focus on the maintenance of appropriate emotional responses and behavior modeling (e.g., no gloating, no sore losers, no emotional aggression or defensiveness).Again, this is all ground-floor stuff, so kids can start learning it as early as 2-ish. Learning to take turns is a huge emotional advantage. As is learning how to both succeed and fail in games and life with grace.
A very good game that has a degree of complexity appropriate to kids that young is "Animal Upon Animal" by HABA.[0]
It's a simple stacking game: On your turn, stack this animal on that animal; if the stack of animals falls over, that's the fail state. (There is a 6-sided dice with pictures on 4 of its faces, and 1 and 2 pips on the the remaining, but it can easily be removed from the game until the child in question is ready for it.)
The 3 year old will tell you what they want to do.
At 3, my little guy's favorite toy was rated for 8+ years old, and his favorite game was chess.
He never cared if he won or lost at chess, his primary purpose was to push his pawns, castle, and then get in at least one en passant.
Maybe start with hopscotch (the regular kind with where you toss an object into a square, but with clearly numbered squares), telling them each time the number of the square they're on, the number of the square they're heading to, and the number of hops they'll do, before they do them. After a while, they'll get an understanding of how the numbers on the squares relate to what you're saying, and maybe even a vague intuition for subtraction.
Eventually, you can replace the thrown object with a rolled die (I suggest a big soccer-ball-sized die they can see from far away, with numbered faces instead of pips)—whose roll indicates how many spaces they should move rather than the space they should move to—and keep playing the same game.
Once they get that, then they're ready to play board games.
My First Carcassonne and Sushi Go! are two we like and they're playable for adults too. (A lot of kids games are terrible, ones based on licenses like Disney or Paw Patrol for example.)
Also check out Peaceable Kingdom they make cooperative games for kids (Hoot Owl Hoot, Race to The Treasure, etc.) I haven't played them personally but based on reviews some families really enjoy them a lot.
https://smile.amazon.com/Jax-8001-Sequence-for-Kids/dp/B0000...