Getting to the Dry Tortugas will be a pleasant downwind sail and there are plenty of anchorages once you are there. It would be a good place to explore for a couple of weeks.
But and it is a big but getting back to Key West will be a 70 mile beat into the wind unless you are very very lucky. I would not want to have to do it on a fixed schedule even on a biggish boat.
A Catalina 22 could make the trip esp. if you waited for a good weather window for the return but you could be waiting for weeks.
You can only stay for 14 days, so waiting for weeks isn't an option.
The main thing is to learn what your doing first. ASA 101 and 103 are the important ones to start out on, personally I don't see the need for others in less you just love getting certs. After that you need to build up as much experience as possible, including many day sails and overnighters closer to land on the boat you'll be taking or a similar one if you're chartering. Also study the weather there, and study weather in general. Watch it for a while to get a clue of the patterns, which will be different during each season. Finally start planning your trip and wait for the ideal pattern. If you time the trip right, you can actually catch favorable wind in each direction, or at least give yourself a really good chance of avoiding dangerous weather. Remember to bring an EPIRB and a backup VHF just in case, as well as all the supplies for your entire trip, there are no provisions there.
I recently sailed from Tampa Bay to Garden Key and then on to Key West and finally Miami. Unfortunately I had a schedule I had to keep to, so we left in somewhat foul weather, but the reward after 200 miles of open water was well worth it. It's quite a place. Keep sure to respect it when you get there and obey the regulations as the rangers there are very strict and the environment is a national treasure.