Go Away, Monster! is the greatest first game I've ever seen. It teaches shape identification by blind touch and eye, eases fears about monsters, and we play with the rule that when choose an object you don't need, you give it to someone else who needs it. So it's not cooperative, but still fairly collegial. Also playable as young as maybe two--by five, you might be a little past the sweet spot.
Dino Hunt Dice and Love Letter have been our go-to quick, portable games, and both come in a variety of themes.
But the best surprise for me was that my son got very interested in Splendor at that age. There's really nothing about it which gates off younger players. He played sort of terribly, mostly just hoarding coins, but it was the first game he didn't mind losing.
Giro Galoppo is a horse-racing game with some planning ahead and yomi elements. My kids got sort of caught up in playing with the pieces almost more than playing the game, but it's great for playing with kids.
Unfortunately, one of our biggest hits was an out-of-print thrift store find, the old Powerpuff Girls: Saving the World Before Bedtime. A surprisingly approachable way to introduce variable turn order and characters who don't represent players, but over whom players have limited control. Cone to think of it, it's probably the perfect way to get your kids ready for Imperial.