The answer really depends upon what kind of vacation you want.
Do you want to spend a lot of time with your kids, or have clubs that are open all day on sea days - so that you have more flexibility with your schedule?
Are you looking for a port intensive itinerary or do you want to stay on the ship most of the time?
Here's what I can tell you first off...
In the Carnival Cruise Lines family, Carnival is the budget line and is more of a party ship (was just on Freedom in December, and NEVER again). Princess is more upscale and you'll find all ages onboard (was on the Crown last June and the Coral earlier this month). Holland America is higher up the scale and usually has an older crowd. I've heard that their kids programs are well run, but you might not find a lot of travelers in your and your kid's ages. Costa is "cruising Italian style" and we had a good experience with their kids club but that was in Europe. I'm not sure how their US cruises are run. Carnival also owns Cunard and the Yachts of Seabourn, but I don't think that's what you're looking for.
Royal Caribbean International has Royal Caribbean Cruise Line. We LOVE their Voyager and Freedom class ships (we were on both last summer). There is so much to do! However, their kids clubs do close from 12-2 and 5-7 each day. And there is an extra charge for late night group babysitting after 10pm. However, you can signup for in cabin babysitting for a very reasonable price. My DD is 10 and these ships are perfect for her. For your kids the Freedom class might be best, as they have the big water park on board. RCI also has Celebrity, which is supposed to have a good kids program but we've never tried. They also have Azmara which is an adult small-ship experience.
Then there's NCL. We had a bad experience on the Star when DD was 4, but we were on the Jade in December and LOVED it! And after 5
DCL cruises, she had a great time in the Kids club on NCL. I do think that the cruise line in general is hit or miss, so I'm not too sure of the other ships, but can definitely recommend the Jade.
You can checkout cruisecriticdotcom for more information on family cruising and the different ships.
Good luck!