The coolest thing to do is go to the Dry Tortugas and Fort Jefferson. It will take an entire day and there are only two real ways to get there, boat or plane. Only 50K people visit the park every year so you would be one of a few.
I'm a cat guy so the Hemingway House is a must every time I go. If you have a slight buzz on the Butterfly Conservatory (or whatever it is called) is a trip. As far as beer, there isn't much to find in terms of craft so I typically start at one end of Duval and head to the other drinking mixed drinks, frozen drinks, mojitos, or whatever is on special.