Key West is a great place to do your own thing. Any particular interests, or just the touristy stuff? Go to Mallory Square and there is a lot of stuff to see and do. Buy an all-day pass on the Old Town Trolley and you can go everywhere it goes, and get on or off when and where you want. Another trolley will be along every so often. Don't confuse this with the Conch Train, which doesn't have the come and go option.
Mel Fisher's museum is a must do for tourists. There is a little aquarium too - across the street. Get some frozen key lime pie on a stick at the Blonde Giraffe. Get a cheeseburger at Margaritaville. You'll be in tourist heaven after all that!
Just be careful with the t-shirt shops and an offer to do a "custom" design. You can easily end up with a $50 t-shirt at some of the shops if you aren't careful.
Museum first, since you will avoid the crush of people going to Margaritaville. Then go to Jimmy's place later in the afternoon. Actually time has little importance in Key West. The whole place seems to be in "whenever" time!
I agree, the Mel Fisher Museum is a must! Especially, if you are into history. I thought it was really cool, to see all the old artifacts. I would go there first, before it gets too crowded. I also was lucky enough to catch a history channel show on Mel Fisher, and his search for the Atocha. It just happen to come on about 1 month before we went. It was perfect timing. Lil' Lori