Hello Meghan and welcome to the forum!
Take the train from Florence to Rome
https://www.trenitalia.com/content/tcom/en.html
Station names:
Firenze SM Novella
Roma Termini
From the station in Rome- take a taxi to your hotel- you’ll need 2 taxis- ( unless there happens to be a taxi van) go outside to the official taxi rank and wait your turn. Have your hotel address on a slip of paper to hand to the driver.
The fare shouldn’t be much more than 10 euros if you are staying in the historic center.
All taxis are required by law to accept credit cards but check with driver to make sure their machine isn’t “broken”
Cash is always fine
Tours- Both Walks of Italy and Tour Guy have Rome in a Day tours that include Vatican, Colosseum, etc. It’s a pretty long day- just be sure you really need all of that!
You can see some of the sights on your own- Trevi, Spanish Steps, Pantheon, etc
The RS Heart of Rome walk is a great first day activity in Rome- safe in the evening and the fountains will all be lit up.
https://www.walksofitaly.com/rome-tours
https://thetourguy.com/tours/rome
Both are mostly well reviewed by members here. As far as hotel pick up- not sure you’ll get that unless you find a private guide- and your hotel will likely be in a pedestrianized area so you'd need to walk out to a pick up point.
Hotels- stay in the historic center near Pantheon/Piazza Navonna/Campo De’Fiori. The area is mostly all pedestrianized and perfectly safe. You can walk to almost all sights from there. Lots of restaurants.
A forum favorite is Hotel Smeraldo.
https://www.smeraldoroma.com/en/
Use booking.com to find more options- use the filters for budget, location, amenities needed like AC, elevator, etc.
You can trust reviews there as they can only be left by folks who have actually stayed at the property.
Departure- taxi to airport is a fixed rate 55 euros but your group would need 2 cars.
Check Rome Cabs (not a taxi- but a driver service) - again highly recommended here- they probably have a van.
https://www.romecabs.com/
Lots of other suggestions here for hotels, etc.
Here is a Search of “Rome hotels” all post from past year:
https://search.ricksteves.com/?button=&date_range=1y&filter=Travel+Forum&query=Rome+hotels&utf8=✓
Enjoy! Rome is our favorite city.
ETA: Many people like to stay near the Termini station- there are several nice hotels there. It's a bit further walk to sights but also close to public transportation. I would still encourage you to stay in the Centro Storico for the best Rome experience.
Hotel Sonya and Hotel Aberdeen are within walking distance of the station. The plus is you don't need a taxi on arrival and you can take the Leonardo Express to FCO (although at 14 euros a person a car service will probably be less expensive for you)