How far away is Sorrento from Rome?

It is about 1 hr. 10 min. from Rome to Naples on the Eurostar (ES) train and then another 30-45 minutes, give or take, to Sorrento after you switch trains. You would definitely want to take the train, and a day trip to Sorrento is definitely doable if you leave early enough to get a good day's worth out of the trip.

The travel time on fast EuroStar Italia trains from Rome to Naples ranges from 1hr10min to 1hr45min. Go here for timetables and fares. If you leave Rome at 06:45, you'll arrive in Naples at 08:30.

The travel time on Circumvesuviana trains from Naples to Sorrento is 1hr7min. See the timetables here. Since you'll need to wait until you get to Naples to buy a ticket for the Circumvesuviana, you won't be able to catch the 08:40 train. If you take the 09:10 train, you'll arrive in Sorrento at 10:17.

Circumvesuviana trains depart from the lower level of the Napoli Centrale station.

Ok, so now what I am thinking for our itinerary is:

4 nights in Rome
then go to Sorrento for 2-3 nights.

While we are in Sorrento, would the Amalfi coast be doable for a day trip from there?

After that, I am thinking Florence for 3 nights (going to Tuscany for a day trip),
then Venice for 2-3 nights.

Does this make sense?

Sorrento is an exellent jumping off point for day trips to Pompeii, Capri, the rest of the Amalfi Coast, and even Naples. I've done your exact itinerary myself and it's very doable.

I love Sorrento and the coast but would want a few nights there. My only thought is what month are you traveling in. It is more seasonal than other parts. The ferry rides to Positano and Capri are beautiful as you look straight up at the towns clinging to the cliffs and that beautiful water below.