Two things to keep in mind in your itinerary is what time of day your flights will be arriving/leaving, and the fact that you will be jet-lagged and weary after a full-day of air travel. Depending on what time of day you arrive, you may want to consider starting your trip in Rome rather than extending your arrival by another couple of hours to get to Florence. The regional train from FCO to Roma Termini is a half-hour ride, so would be able to get to your hotel, settle in and start unwinding/exploring fairly quickly after dealing with flying, customs, etc. Likewise, if your departure flight is in the morning or early afternoon, then it makes sense to visit Florence second, since the last day will be lost to travel anyway
However, if you do decide to go directly to Florence, their are two direct trains from FCO to Florence, which leave at 11am & 3pm. More info here:
https://www.italiarail.com/pages/routes/rome-fiumicino-airport-to-florence
You can also take the regional train from the airport to Termini or Tiburtina stations in Rome, then transfer to a high-speed train to Florence. This is more of a hassle but gives you more flexibility with travel times.
As for the cities themselves, Florence is a very walkable city, so anywhere in the central area between Santa Maria Novella, Sant Ambroggio, and the Arno River is great. For a bit less crowds, check out Oltrarno (south side of the Arno). As far as day trips go, it depends on what you are seeking. If you are into wine tasting, there are plenty of Chianti day trips that travel to a winery or two and one of the towns in the Chianti region such as Greve. If you just want a solid day trip, I would recommend the Walkabout "Best of Tuscany" Tour, which takes you to Siena, then to an organic farm in the countryside for lunch, then to San Gimignano (a scenic hilltown famous for its many towers), and finishes off at the Piazza del Duomo in Pisa (aka home of the Leaning Tower). My wife and I did that during my 4-day stay in Florence a few years ago and we had a great time.
Rome: Since you have a relatively short stay you will want to be in the middle of the action. Booking anywhere near Campo de Fiori, Piazza Navona, or Pantheon would put you in the heart of the old city. Trastevere is another great neighborhood w/ good nightlife. My favorite stay was just off the Campo de Fiori, which is a cool little piazza which hosts a farmers market every morning. You may want to check Airbnb as there's lots of apartments for relatively cheap, especially since you are traveling solo. As for sights, what's must-do depends on where your interests lie. The most famous sights (Coliseum/Forum, Pantheon, Vatican, etc.) are all "can't miss" in my opinion, but my favorite part of Rome is just exploring the city center, as it is a messy, wonderful, vibrant maze to get lost in. Check out Rick Steves' "Heart of Rome" walk for a more structured version of this:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=2ahUKEwiOg979-uHeAhXpjlQKHeLuAo4QFjAAegQIExAC&url=https%3A%2F%2Fwww.ricksteves.com%2Fwatch-read-listen%2Faudio%2Faudio-tours%2Fheart-of-rome-walk-map-1.pdf&usg=AOvVaw2F_nzksX__ufKOFxsigtkH