I got Hoot Owl Hoot for my 3 year old Niece this year. It is a super easy game and since the game is not long it keeps the kids attention.
My niece also has Barnyard Uno which is a great game for younger ones. Very easy and help teach colors.
Chutes & Ladders, Candyland, Hungry Hungry Hippos, Hi Ho Cherry Oh, Connect Four, and not technically a board game, but putting together a puzzle is always good as well. These are a lot of games that I played when I was about that age anyway. I always enjoyed them. 🙂
Candyland is great to start-off with. My 4yr old boy enjoys playing sequence – the animals version. It’s very easy to play but teaches you strategy and critical thinking. He also loves any kind / variation of bingo.