Hi, I am new here but thought I would chime in. We LOVE games at our house, and my DH has amassed quite a collection of really great games by reading boardgamegeek.com. It has useful reviews and also, for some games, instructions on how to handicap the older players to even the playing field.
Another tip that we used with our DD5 when she was younger is to ask her if she would like us to play "easy, medium, or hard" and adjust our strategies accordingly. Obviously this doesn't work as well with older siblings.
We only buy games that will be enjoyable for the adults as well as the child. We got rid of the mind-numbing candyland type games (ugh!).
Here are some game suggestions that we have been playing with our DD since she was 3.
Labyrinth (skip the junior version, give the experienced players more cards to find or let junior players PICK which card to find next)
Blink
Blokus, DD also likes to play this symmetrically where she mirrors what my DH does.
Pick Picnic, highly recommended
Castle keep, has lots of optional rules
Rat a tat Cat
Doodle dice, very fun!
Sequence for kids
Skip bo junior
Chicken cha-cha-cha (the best memory game I've ever played)
Incan Gold
Giro Galoppo
Marrakesh
Some of these games can be hard to locate, so here are some sites that carry them: timewellspent.org, funagain.com, and boardgameprices.com
Hope that helps!
Dawn