Although I used to suggest new cruisers stick with the ship's tours on that first cruise, looking at your ports you will be fine doing tours privately. Is your cruise the 4 or 5 night? Cozumel, Key West, and Grand Cayman are all perfectly safe places to set things up on your own.
You can arrange something on the pier in Cozumel and Grand Cayman. There are always tons of vans and taxis waiting where you can do a tour exactly like those offered by the ship, or customize one. Key West can be a different scenario, depending on where you dock. However there are lots of options in and around Mallory Square. It is a small town and you can walk around to see the various things, or take the Old Town Trolley where you can hop on and off at your leisure. There is also the Conch Train but it doesn't let you come and go as you please. Just google Key West Tours and you'll see some good info.
If you tell us more about what you want to do we can give you more specific advice.
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