Hi Michelle,
Welcome to the Moms Panel, and congratulations on booking your first Disney Cruise! By now I am guessing your travel documents may have arrived and the excitement is really beginning to build!
Key West is an amazing port, and a real treat because it is not one of the standard places that Disney Cruise Line visits regularly. There are plenty of Port Adventures available through Disney Cruise Line, but I also believe that this is one port that offers a quaint, easy to navigate city that is quite possibly simple enough to do on your own. Certainly I will caution you to keep in mind that when exploring or booking independent excursions on your own - it is your responsibility to make it back to the ship on time.
You didn't mention the ages of your children, but I recently offered some advice
HERE about visiting with nine year olds. I think the fishing charter sounds like a lot of fun, just be sure to verify the cancellation policies in the event of bad weather and the departure/arrival times so you can make sure you have ample time to get back to the ship.
One other opportunity I've heard a lot about, but haven't tried is the electric car rentals. You can explore the island on your own, and they'll even pick you up at the pier. One of the popular companies is Pirate Scooter Rentals, at quick internet search will turn up plenty of information!
I hope this helps, have a great time!