I can remember getting started on Monopoly around that time (7).

I think you've got a great idea there.  I like Family Game night on a number of different levels.  A year or so ago I wrote this little essay for the Prevent Child Abuse MN newsletter.  It got picked up by a couple of websites and was on my blog before it became defunct.  Here's a link.

Winter Got You Stuck Inside? Turn off the TV and Bring out the Family Games

We like Connect Four and Dominoes as well. 

Others I'd recommend follow here: 
The Uncle Wiggily Game (2-4 players, ages 4-8)
The I Spy Memory Game (1-6 players, 5+)
Operation (1+ players, ages 6+)
Mille Bornes (2-6 players, ages 8+)
Sorry (2-4 players, 6+) 

Sorry remains our family favorite. We even keep an on-going tally of wins.
 
As for the adults, my personal favorites are Taboo, Guesstures, Scattergories, Hearts, Spades, and Dominoes.

I'd recommend Apples to Apples (family or kids versions). It's a great social game that kids can easily pick up. Family Fluxx and Aquarius by Looney Labs games are good.

Check out children's games/family games sections on BoardGameGeek.com. It's a great way to find new games and to get an idea of how fun they are. There are many kids games out there that are fun for kids, and an interesting/good game at the same time.

Blokus is good as a family game, and Connect 4 is easy for kids to play together.

Blokus and Apples to Apples.
Mancala is good for 2. I don't know if there's a 4-player variation.

I just played a great game that would fit the first two, and probably all three categories.  It is called "In a Pickle" and it is a word-based card game that plays fast and is enjoyable for all ages.  We had a competitive group of late teens and adults, all laughing and enjoying it.  My five year old joined us and won the game.  He didn't need any help with the rules although we did define a couple words for him.  So it is educational too ;-).

Blokus can get intense, which is why we say "Blokus!" as if we're cussing when we play it.
I like Master Mind, in part because if you play it systematically, you'll win every time. It's a good teaching game for kids to learn logic.
We play Catan with a 6-year-old, but my 12-year-old noticed that I favor the young one when we're playing. I make trades with her that help her, put my roads where they don't block her, and that tips the scales enough against everyone else that she'll win. I might have to stop doing that.
I'll repeat the Apples to Apples recommendation, but suggest the kids' version. It's not that the adult version is inappropriate, it's just that kids won't know who the people are on too many of the cards. There's too much for them not to understand.
I also like chess, Othello, Stratego, Monopoly if you have time to kill. Snap it Up and Blink are fun, quick card games (but stressful if you don't like to play quickly). Loot is a pirate card game my son likes.
I don't like Sorry, too heartbreaking. I've never understood Life very well; is there strategy? And I can't believe I recently let Candyland back into the house, which, like baseball, could theoretically never end.

I really like "The Scrambled States of America" for grade school age kids.  It's a card game that teaches them states and it's fun enough and varied enough for kids of different ages to play together and also fun for adults (or at least tolerable)

"Clue" is always good.  I agree with the "Mancala" and "Apples to Apples" suggestions.

There is always poker and black jack too.  Get a set of chips and cards and there is always fun to be had.

There is a fun, unique board game called "Herd Your Horses" which is a hit with girls of all ages.  It's got a variety of ways to play and if your daughter is horse crazy all the better.

One of my goslings is getting a bit old for candy land but she still loves it and she has taken cards from 3 or more garage sale Candy Land games and put them together.  If you cull out some of the color cards and leave in the characters then the game jumps around a lot more at a faster pace (a la Sorry) and it is more interesting. Downside ... it may make it even longer but it can make it shorter, too.

As a multiple-time champion and member of the Twin Ports "Evil UNO" professional UNO league I have to say that I second TopOfTheHillMan's suggestion.