This was me before our first cruise that we took just last week. We did the same cruise you are doing and loved it. Never thought I would enjoy cruising, but it was just a really cool experience.
Most people have already answered your questions, but as a newbie that regularly goes to the parks, I can share my thoughts. We did get off at Nassau, mainly because we felt we should set foot there at least once. Our kids are 12 and 7 so a bit older than yours, but they found it interesting just because it was so different from anything else they have seen. We went in some shops, Straw Market then walked to the beach area. I don't think you are missing much if you don't get off, but I would say try it out. If I did another cruise that stopped there, I would probably skip it that time. The good thing is that you can walk back to the ship whenever you want. We came back for lunch on the ship so we weren't there a super long time. Make sure that when you leave and return to the ship for Nassau or CC that each traveler has their KTTW card and 18+ will need a govt issued ID (either Driver's license or passport.)
Agree with others that for your first cruise, I think you are better off not booking a ton of extras. We only booked swimming with stingrays at
Castaway Cay, which I highly recommend, but maybe when your kids are a little older. My husband and daughter also did the 5k on Castaway Cay and you get a nice souvenir after. It doesn't require any advanced registration and you just do it on your own (following signs around the island.) We did not book any extras on the ship and I don't regret that decision. We had the most fun just walking around and exploring things.
Everyone says it, but you really don't need to pack as much as you think you do. We had way too much stuff and the rooms are very small with not much drawer space. We had come for WDW right before so that was some of the reason we had too much.
Make sure you search for your cruise group on FB and join if that's something that interests you. I found a lot of the questions asked and answered were helpful to me. Someone organizes the FE/other exchanges groups there that you can sign up to be in if desired (we didn't.)
One thing that no one ever mentioned to me, but it reminded me a bit of being on an airplane. The motion was similar (except for the last night when there was some rough waters) and the toilets, boarding, etc. all sort of gave a feeling of being on an airplane. My daughter got motion sick the first evening. I never had to give her medicine, but I had ordered some acupressure bands for kids and they worked really well. She never had an issue after the first night. You get used to the motion pretty quickly, but it is a bizarre feeling if you have never experienced it.
Review the list of prohibited items and double check suitcases for anything maybe leftover from previous trips. My husband's suitcase was delivered very late because it had to go though extra security and had a small pocket knife/toolset removed that he forgot was in his bag. We thought it was lost so double check your bags for those items and don't be like my husband.

Don't judge anything by the first day. It's chaotic and you will feel like cattle. Do make sure to attend the Sail Away Party - it's a great way to start the trip.
I'm sure I have a lot more thoughts, but feel free to PM me if you have specific questions!