We are rethinking this—Airlines for to Rome and out of Florence

You can check the airlines flying from San Francisco to Rome and from Florence to San Francisco at www.Skyscanner.com

ITA Airlines flies non-stop to Rome. They are partnered with KLM who flies out of Florence and can make a connection in Amsterdam for a non-stop to San Francisco..

The best options out of Florence to SFO are via the following European gateways.
1. Via Frankfurt or Munich (Via AirDolomiti, part of Lufthansa Group) - Star Alliance
2. Via Zurich (Via Swiss, part of Lufthansa Group) - Star Alliance
3. Via Amsterdam (Via KLM, part of AirFrance-KLM Group) - Sky Team Alliance
4. Via Paris CDG (Via Air France) - Sly Team Alliance

Lufthansa Group is part of the Star Alliance, of which United is part. Therefore if you fly combinations 1 or 2 above, it is likely a code share with United, which also operates direct flights from the German getaways above to SFO.
Air France-KLM Group is part of the SkyTeam alliance, of which Delta is part, therefore if you fly combinations 3 or 4 it is likely a code share flight with Delta as well.

You can go to Rome via the above gateways with the same airline combinations. In addition SFO has a non stop flight to Rome with ITA Airways (part of the SkyTeam alliance) and one with United (part of the Star Alliance).

Many are fans of open jaw tickets, I do it myself on this next trip, into Lisbon, out of Amsterdam.

But do consider the whole logistics of the flights. If they can get a direct US/Rome flight and a direct Florence/US flight, great. If costs start to add up, or to do so means extra connections and layovers, or a 5 AM flight, left wondering how to get to the airport at 3AM, then a fast train in 1 and a half hours back to Rome starts to make a lot of sense, even if you plan your last night in Rome.

It makes more sense to fly into Florence and home from Rome. More options especially for the return flight. As far as I am concerned, all the airlines are about the same so I would select schedule and price in that order especially if you are looking at next May.

Unless you fly direct FCO-SFO with United or ITA, you are going to be getting up early.

Of course from Rome you have the option of flying with later flights if you return via East Coast airports (EWR, JFK, IAD, PHL, ATL), which depart later in the morning, but which I avoid like the plague, because flying via those airports means prolonging the journey by several hours, in addition to having to do customs at those first ports of entry, before having to check in your luggage again to SFO. The fastest option is to fly directly from a European gateways to SFO via the Artic route. That has the added benefit that SFO will be your first port of entry in the US, which means that once you do customs you don't need to recheck luggage and catch another flight for another 6+ hours. Once in SFO you are home.

Florence of course has fewer options compared to Rome, but some Florence options are not that early in the morning. For example if you come to SFO via Munich or Paris (with Lufthansa or AirFrance respectively) the flights out of Florence depart after 9am, and you just need to be at the airport at around 8am, not a difficult feat since Florence airport is only 15 min from the city center (3+ miles) and being so small you can check in just one hour prior to departure, whereas the Rome airport is 30+ km from the city center (30-40 min) and being very large you should try to be there about 2 hours prior to departure even for a European flight.

Roberto pretty much laid it out. SFO is a United hub, and being the flagship carrier of Star Alliance, you're going to have the most flight options with UAL and its code sharing airlines. American Airline has a fair number of flights also since they moved into the new Terminal-1, they're apart of oneworld Alliance, which includes British Airways. I utilize Google Flights to get an idea of costs, and bookmark those flight-legs that I've got my eye-on.

SFO to FCO direct I believe there's one flight per day, all other flights you're going to change planes in Newark, Chicago, Dulles domestically; overseas changing planes will be in Frankfurt, Munich, Zurich, Toronto and Heathrow.

Returning back to SFO from Florence, you can either fly out of FLR (Florence) or, the more busier airport out of PSA (Pisa). Flying out of either will be a regional flight with a plane change at a much bigger hub like Milan, Frankfurt or, Heathrow.