I just got back froma Princess cruise and loved it! We had a very bad experience on Norwegian and would never sail with them again. I thinl Royal Carribean is best for families , but Princess is a very close second. Book with Princess! You won't be disappointed.
We, on the other hand, love NCL. Especially when you're traveling with a wide variety of ages, the freestyle dining concept works out great. No assigned tablemates, no assigned seating time. I didn't cruise until my kids were too old for the teen club, but most of the reviews of their kids programs seem to be good.
We'll be sailing on the Jewel in November, so I can't give you any first-hand knowledge, but our friends who traveled on the Jewel last year loved it.
I had a co-worker whose first cruise was on Princess, and she didn't enjoy it very much. She thought the staff was rude and not at all friendly. Their cruise companions had been on NCL and RCCL, and said they'd never cruise Princess again, as the staff on the other 2 lines was so much better.
You can find positives or negatives about all the lines, and they definitely have their own personalities. Hopefully you can find the one that works best for you and your family.