welcome ! you should love your first cruise.
1) I always take the cruise-line shuttle. Can't help you on private shuttles.
2) In Nassau the pirate museum would be a hit with most kids. The tour I took also included 2 forts.
In St Thomas there are many options as well. If US national parks are your thing then a tour of St John is great. Kids should enjoy Coral World or Atlantis submarine.
3) any advice? Where to start? tips and soda aren't included. Formal nights have can't miss shows. If you think you might be prone to sea sickness, talk to your doctor about "the patch". Take Dramamine even if you aren't prone to motion sickness...it helps me get my sea legs.