Sorry I did not take the time to read all the previous posts. I have been on 5 different cruiselines and many cruises. For your particular cruise I would not use Holland America. They cater to an older crowd (more like my age) and would not have the activities you would want for your children.
While Disney is lovely, I don't think I would do it for a first cruise.
I took my whole family, 12 people, 7 adults and 5 children ranging in age from 7 to 20 on a Thanksgiving cruise in 2008. We went on Royal Caribbean Adventure of the Seas out of Puerto Rico. We did the southern caribbean going to Aruba, Curacao, St. Maarten,, St. Thomas and I forget, there on more island.
I suggest this itinerary and ship for the following reasons:
2 days at sea offer an opportunity to enjoy the ship which has rock climbing, ice skating, basketball courts, and more more more. They have an excellent children's program. The 13 year old boy was not going to go to the kids things and my son told him you have to try it once and if you don't like it you don't have to go again. He went, made a bunch of friends and loved it. They have a lot of age appopriate things, like wi fi, and more.
For a first cruise, i's not as elegant as some lines, but the ships themselves are very, very pretty, and offer lots to do.
Additionally those islands are the nicest one offered in the caribbean.
The western caribbean is not as nice in my opinion, but more historical if that's what you are looking for. Mayan ruins and such.