You'll take regional trains Rome-Viterbo and don't have to buy the ticket in advance in order to lock in a low fare. Allow time at the train station for slow-moving ticket lines.
Connections to Florence are more frequent, but that trip will take 3-1/2 to 4 hours. A trip partly on the fast Freccia train may save a bit of time but may cost considerably more. Buying early to get a bargain ticket would lock you into a specific departure time (no refunds on promo fares).
You can look at schedules on the user-friendly Deutsche Bahn website. To buy tickets in advance (which you'll want to do for Rome-Florence or Florence-Rome, where there is much money to be saved), use Trenitalia.com, trainline.eu or loco2.com.
Viterbo has a walled city center yet is remarkably non-touristy. The historic core is very atmospheric and a great place to wander around, free of other foreign tourists. A few years ago the tourist office was friendly but a bit, umm, clueless, lacking the ability to tell me where I could catch the bus out to the Villa Lante, which is one of the top sights in the area (for its gardens; not necessarily a good destination in January). I don't remember whether English was spoken; it probably was, but don't expect that at places like the bus station and train station.
I'd suggest printing out a map of Viterbo's historic core from Google Maps so you can get yourself around if the tourist office happens not to be open when you arrive. It's possible it will not have really long hours, or be open 7 days a week, especially in mid-winter. Mark your map with points of interest in case there's not much information at your hotel (which probably will be able to give you a map).
Viterbo isn't a particularly large city, but unless something has changed since 2015, it uses two train stations. Traveling to/from Rome and Florence, you will be using Porta Fiorentina, which is just east of the walled center. I see that most of the hotels are clustered to the south side (many less than 1/2 mile from the station), but there is one hotel, the Balletti Palace, quite near the Porta Fiorentina station. You can find a lot of reviews on booking.com.
The average high in Viterbo will be about 50F, which means some days may be rather cold, especially when you head out in the morning and after the sun goes down. I agree that with time so short and January weather, Venice is a candidate for being skipped this time around. You would not have time to do it justice, anyway.
I haven't done the high-traffic Roman sights recently, but people speak of needing to take line-avoidance action for these:
- The Vatican Museums are mobbed, with most people recommending an early-access tour. Refer to the "Out of hours" options on the ticket website.
- The Vatican Scavi tour sells out very early
- The Colosseum is busy; book especially early if you want more than the standard visit. The official ticket agent is Coop Culture.
- Domus Aurea tickets are snapped up far in advance.
- The Borghese Gallery requires pre-purchased tickets and sells out early.
The Roma Pass might be worthwhile.