I'm quite familiar with that area, and have done a day trip from Florence. Given the transportation time, my suggestion would be to focus on the five villages of the Cinque Terre as that will be easier to accomplish in the time you have. The towns have an interesting history and are somewhat unique.
if you want to spend a short time in each of the towns, you could start in Riomaggiore and work your way north using the frequent local trains. Finish the day with a fine meal in one of the nice restaurants in Monterosso, which is the largest of the five towns, and take the train back to Florence from there. If you decide to do something like that, be sure to check the train times on the website that was suggested in the previous reply.
You could take a stroll on the famous Via dell' Amore, which is a trail that runs from Riomaggiore to Manarola. I believe reservations are required. There's probably information on that in the Rick Steves Italy guidebook.
If you like sweet dessert wines, check out the local Sciacchetrà in the Cinque Terre.
Portovenere is nice but as mentioned, it can only be reached by bus from La Spezia or the local ferries, which may or may not be running depending on the weather. I found it less interesting than the Cinque Terre.
If you haven't used the trains in Italy before, be sure to do some research on the "rules", as there are some potentially expensive caveats to be aware of.