Congrats SnowWitch!!
KaterKat, part of the Disney trip we took when Madison was 3 months old was a
Disney cruise. She was 15 weeks, so just above the minimum age of 12 weeks (she got a certificate that said she was the youngest cruiser!). We had the trip planned about 6 months before we found out about her, so we were really, really hoping she'd be born on time or we'd have to cancel (luckily she was 2 weeks early).
Anyway, it was a lot of fun, and really pretty easy for us since everything was right there-food, pool, room, etc. Easier than Disney was because there was lots to do without walking around in the heat all day.
We also went on a cruise when she was 10 months old, on Princess cruise line, also to the Caribbean. Also a great trip. We had to visit the ship dr for her then cause she had this nasty cold that would not die, and she'd cough so hard she'd vomit everynight. Her own dr said it would go away on its own before we left, but it seemed to be getting worse. They didn't really have much medication for infants, and we called her dr to check on what they gave us and they said don't use it, so just a piece of advice if you happen to need the ship dr for your little guy.
On both we found enough shore excursions she could participate in (though, yeah, the really cool ones didn't allow babies) and both ships had tons of kids on board.
We did an Alaskan cruise on Celebrity when I was pregnant and it was beautiful so if you can, definitely go. I'd love to go back when I can experience it without the morning sickness. They were the best cruise line, in our opinion.