Did this while visiting. They have a variety of canoe trips on the Hillsborough River. I did "Trip #1" which involved two hours of unguided canoeing without seeing any signs of civilization. I met two kayakers on the river and that was it. Saw tons of water birds, alligators, and turtles and had a great time.
When I went, the water was low, so it was a little hard to maneuver the canoe in places. I got stuck briefly a few times, but never badly enough that I had to get out of the boat. Might be on the adventurous side for some.
Almost all of the alligators I saw were on "Flint Creek" which is the side trip to the right just after you put in. If you are afraid of alligators, just skip that part, you will see few to none on the main river. It didn't seem dangerous to me, but I can easily imagine a few of my friends having panic attacks.
They put you in the water and pick you up at the other end so you just hop in and paddle. Fun for tourists and maybe for the locals too. Rent the back supports and the dry bag, they are worth it.
Short drive from Tampa. The traffic coming back was a disaster at rush hour, so plan accordingly.