Not only have I cruised to Bermuda, I lived there for five years when I was younger (military father, used to be a base on the island...just dumb lucky).
Bermuda is one of my favourite places in the world. So much so that I had my wedding there three years ago...that's when we cruised.
I'll try not to let myself get carried away, but there is a lot to see and do in spite of the island being so small. Definitely see the South Shore beaches (Horseshoe Bay, Warwick Long Bay, Stonehole Bay are three of my favourites), Gibb's Hill Lighthouse, Crystal and/or Fantasy Caves, The Botanical Gardens. Take some time to walk around the capital of Hamilton, there's some construction on Front Street, but walking the laneways, streets and parks makes for a nice afternoon. Walking around St. George's is also amazing, it's the oldest part of the island (and a World Heritage Site), and you can just feel the history in the buildings and the laneways. My biggest reccommendation would be to take a ferry ride around the Sound, as close to sunset as possible. It is just a really great way to experience the beauty of the island.
For saving money, get the transportation pass and use the buses and ferries as much as possible. It's cheaper than renting a moped (and safer) or taking a taxi, and the transportation system is pretty great. Just remember to say "hello" or "good day" to the bus drivers, and if you are unsure where to go ask them, they are more than willing to help. Don't eat out at too many restaurants, they are pricey. But pick at least one splurge meal, you won't be sorry. If you like Thai food there is a place called "Silk" on Front Street in Hamilton that is amazing. Or if you like Italian, La Trattoria (also in Hamilton) is my all time favourite (we had our wedding dinner here). Shopping is fairly pricey as well, but you can get some good deals on linens and crystal. Don't go expecting to find bargains or bartering, you won't find them.
oooh...I let myself get carried away. OOps! Where is your ship docking? How long are you docked?
I think I should stop there for now...feel free to ask any questions though! I'll do my best to answer them.
Have a great cruise!