Do I need to make reservations in advance for trains from Cinque Terra to Firenze?

That is the normal route. The fastest connections will be suggested when you search schedules at http://reiseauskunft.bahn.de/bin/query.exe/en or at www.trenitalia.com. You absolutely don't need to buy tickets ahead. Most trains operating are regional trains and those are cheap, nonreserved tickets. If you buy them at a train station anywhere in Italy, then you date stamp them in the station on the way to the platform. Don't get on a faster, reserved InterCity train by accident. Regional train tickets bought online have more restrictive timing and are not cheaper.

Thank you; that helps.

I doubt that you'd save much with advance purchase tickets on that route, so the easiest solution is just to buy tickets in the Cinque Terre, preferably a day or two before you'll be travelling. Before buying tickets, it's a good idea to research the various solutions using the websites that Laura provided, so that you can choose the quickest and easiest one (shortest time, fewest changes).

Which of the five Cinque Terre towns are you staying in?

One important point to mention is that you MUST validate (time & date stamp) locally purchased Regionale tickets prior to boarding the train on the day of travel, or you may face hefty fines which will be collected on the spot.

Thanks, Ken (and Laura)
We're staying in Vernazza.
Do I get the ticket validated at the train station right before I leave?

Yes, just punch it in the device when you arrive in the station. Once a regional train ticket is stamped, you only have a few hours to complete the ride, with the allowed window depending on distance. If you have Rick's Italy book, see also Getting Around the Cinque Terre on p. 346.