My kids (3 and 5), but particularly my 5-year-old, LOVE games. Here are some of their favorites:
Kids of Carcassonne
Tier Auf Tier
Socken Zocken
Trouble
Guess Who
Chess (the 5-year-old; we haven't taught the 3-year-old)
Blokus
Ker-plunk
A current favorite with them is a card game called Sleeping Queens.
And their grandparents brought Uno Moo with them when they visited last weekend. The kids seemed to really like it, but I didn't play with them, so I don't really know anything about it--except that they loved it.
Of those that I listed, my 3-and-5-year-olds can set up and play all but Chess independently (I should probably say, my 5-year-old can set them up and lead my 3-year-old through them).