HI Erika
Here are my two cents - we took our then-16 mth old on a RCCL cruise in November. We all had a ball and our daughter had a wonderful time.
We did do an exscursion to the Mayan Ruins in Belize - my husband "wore" her in a baby backpack which worked great. He also wore this around the ship and it worked well. I think shore exscurions are fine but can be expensive - we had to pay about 80% of the price for my daughter to go. Some are free for young children. I would just try to schedule a morning one if your baby takes an afternoon nap, etc. We booked ours online before we went so we didn't have to wait in the shore exscursion desk lines - which were LONG.
We also did something in every port - like going to the beach, etc. It is nice to get off the ship and see the sights.
We did bring a car seat and regretted it. We tried to use it on the plane but our daughter refused to sit in her seat - only wanted to stand on the floor or sit in our lap. We never used it after that. We just held her in cabs, etc. We also brought a lightweigt stroller, we probably could have lived without this because of the backpack, but my daughter is not a big stroller rider, likes to be walking or in the backpack. I did use it when my husband was not with us. So I would say to bring it - I saw lots of younger babies in their stroller in the dining areas and around the ship.
Other thing I would bring is get the inflateable tub (Snug Tub at Babies R Us) for your room if you don't have a suite with a tub - it works like a charm on the floor of the shower and makes your baby feel just like they are in a normal tub. You could also bring it up to the pool to use by your chair since your baby will not be allowed in the pool.
I am not sure if Celebrity has babysitting - RCCL does and it was WONDERFUL - we had the same babysitter every night from 8pm - 12 am which gave us the opportunity to have wonderful romantic dinners (we had late seating and fed her early in the casual dining areas), and our daughter goes to bed at 8pm so we never really missed any time with her. This worked perfectly, I just did not want to struggle with her in a formal dining setting and she would have never been able to sit in a highchair for 2+ hours. (20 minutes is her limit!

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Other things that I am glad we brought - a bunch of small garbage bags for diapers (your cabin steward will appreciate this!) a bunch of small plastic bags for snacks to carry with you, small toys, etc.
Have a wonderful cruise!
Kara