I have a 9, 6, and 4 year old. They all shoot, and its a blast to do as a family. Below is what we did. So far so good. Only real advice is to get the lightest overall weight you can, as the actual weight of the bow is tougher on the kiddos than pulling it back. At least in my limited experience....
We scored a Cabela's bargain cave deal on a Bear Apprentice for my daughter. She loves the pink camo!
http://www.beararchery.com/bows/compound/apprentice-iii
If you do some research via the net you find a writeup someone did where you can turn the down the draw weight by utilizing extra long limb bolts. Makes it pretty manageable for my daughter.
Cabela's has these on sale right now.
http://goo.gl/dKA5Gz
My son started out on a Fuse Freestyle Bow. Its been a great bow for us and is now the bow my 4 year old shoots. My friend has a 11 year old who still shoots this same bow. He is probably due for an upgrade, but it is a versatile bow (10-25lb draw weight).
http://www.fusearchery.com/youth/
The rest and arrows are total junk. Upgraded to a Whisker Biscuit and some better arrows from the local shop.
We upgraded the oldest to a Mission Menace. Other than being a bit heavy for him, its been a great bow.
The Bear kit options are usually a decent set up. The sight and whisker biscuit rest are light and cheap, great for kids. The arrows suck, just throw them away and forget you ever got them if you go that route.
One of the harder things to chase down is a youth release. The 4 year old finger shoots, but the older two use releases. Had to search around a bit to find one that fits a tiny wrist!
This one works ok:
http://www.cabelas.com/product/scott-archery-hero-youth-release/1507609.uts
We won this bow this bow at the Alaska Bow Hunter's Association Banquet.
http://www.cabelas.com/product/bear-archery-youth-brave-black-compound-bow-package/1769654.uts
It's VERY basic and a big step down from all of the above. Best for a kid just wanting to start out. If anyone has a youngster looking to experiment with a bow in the garage I'll let you have it. Mine is bright green and I think I threw away the arrows. (Will check if someone takes me up on this) Prefer someone in Anchorage to pick it up, but if you are willing to pay the shipping I'll pack it up and tell you the weight, you print a label to PDF and email it to me and I'll mail it to you.
Good luck in your search! Its pretty fun to do with kids.