Unless you are simply crazy for fish, sharks, dolphins and aquariums in general, the Genoa Aquarium is probably a little too far to do for a day trip. That being said, it is very good and you can see the whole thing in about 2 hours - so it is not undoable from Florence. However, if you are in or around Genoa, this is a great thing to do, especially for kids. We stopped there today on our way to France (road trip) and had a fantastic visit.
I really enjoyed the difference between the big strong things (Sharks!) and the small fragile ones, like the anemone below (which is viewed and photographed through a magnifying close built into the case of the tank):