Welcome aboard, Caletta! Congratulations on your new little one! You have chosen the right cruise line for children of all ages.
There are many Broadway-style shows onboard every Disney cruise, and you are certainly welcome to bring your baby to the show with you. I see babies with their families all over the ship (except, of course, in the adults-only areas), so please don't worry about anything!
Occasionally we've heard little ones crying during a show, but their parents usually just quietly excuse themselves out of the theatre and no one seems to mind at all. Babies cry. People understand that. Also, you should know that there are always at least two show times for each show, so you can choose one that works best for your baby's schedule, and plan your dinner seating around that as well.
Be sure to check out the nurseries on Disney Cruise Line's website.
It's A Small World can be found on the Dream and Fantasy.
Flounder's Reef is on the Wonder and Magic. You'd need to make advanced reservations and there is an hourly fee, but you might consider taking your little one there while you enjoy some adult-time too.
Have a great time planning your cruise, and be sure to stop back here and ask any other questions you might have!