I would recommend Cartagena as a game that has easy rules (takes 10 minutes to learn) and still retains strategy. This is my favorite overall game to suggest because it's easy to explain, easy to setup, and you can give kids a concession (you play with an extra man, or they play with one less). Can be played by kids who don't read.
Carcassone is good, fairly straightforward and can be played by kids who can't read.
Settlers is good, but has more complicated rules. Settlers Cities and Knights is also good, but I don't necessarily recommend the other expansions except the 5-6 player one if you're playing with another family.
If you can find Clue Museum Caper, I recommend it, but last I checked, it was discontinued.
Kill Dr. Lucky is fun but has a bit of dark humor, and is better with 4 people.
Pandemic is a good cooperative game (i.e. you work together to win the game and can play open-handed), but has complicated setup. It can be played with 2 players or even solo since you can use as many or as few characters as you want. You can easily adjust the difficulty of the game. I'd recommend it after trying Settlers. I didn't think Forbidden Island (same author) was as good, but it has the same mechanics.
We also like Dominion which can also be played with just two players without losing the interesting aspects. This is a deck building game, so it may come with around 20 cards, but you select any 10 and then play a game--so there's a lot of variety game to game.
If you're ok with cartoon pictures with cleavage and humor in attacking people, Munchkin (Steve Jackson Games) is fun.
For card games, people like Fluxx. Don't own it myself, but have enjoyed playing it.