Hi Eliza,
On my trips to Rome, I stayed in the Termini neighborhood (I will do so again.)
I was in a very nice, quiet hotel. I enjoyed the walks to the attractions and to other places I wandered. Rome is an outside walking museum. Around every corner seems to be a little surprise with architecture, scenery, people, cafe and pastry shops, and little markets.
(The Beehive Hostel gets a lot of good reviews. I never stayed there nor do I know anyone who has.)
Going and coming at different hours of the day from the area, sometimes walking through the Termini bus depot late if I rode a bus, gave me no problems. There are many travelers and locals out and about. Of course, a bit of common sense prevails -- do not venture out very late -- say after 11:00 PM or so. Still, I think you would be okay since a lot of restaurants will be closing up and patrons will be going back to hotels or homes.
Stay on the main drag avoiding alleys or very dark streets.
Although, my first trip to Rome, one night, at the beginning of my trip, I got turned around, ended up in a neighborhood. I happened upon this military vehicle sort of hidden at the end of another street (like a dead end if you will.) I approached them carefully with a smile making eye contact and greeted them in Italian. I asked them for directions back to my hotel. I showed them my hotel card. They got me on my way.
As I walking, I heard the vehicle behind me. They slowed down for a second or two, nodded, and went on their way. My point? You will always find someone to help you. That has been my experience for a thing or two.
I don't make it obvious, but I "scan" the Termini walking area when I get off the bus (or train.) There were a few homeless people sitting on the roof of the entrance to one of the trains. They were drinking. Police were right there though. Still, I avoided walking past them.
There are plenty of police and military surrounding the Termini area. And on the main drags, police cars are always going back and forth. That does not mean trouble or cause to be worried.
Practice a big city awareness -- something I would also advise if you were to visit Boston.
I also carry a purse-size LED flashlight I got at Walmart for a $1. I figured what the heck. It turned out to be a good investment, lol. It is very light-weight and packed a "big" light. It came in handy on those dark December nights. Sometimes cobblestone can be protruding or curbs can be jagged and broken or sidewalks can "just end."
Perhaps use a cross body purse for your daily valuables; a money belt for extra money, cards, Passport if you will be out and about for most of the day.
If your choice is to use a backpack, just secure your money, etc. deep down into the pack and zip it or buckle it.
If you will be riding the bus or train, have your ticket(s) at the ready so you don't have to fiddle with your bag on the bus or the metro station.
With regard to bus tickets, make sure you validate your ticket as soon as you step on the bus. Transit Inspectors are increasing their presence on major bus routes.
If your ticket is not validated or you don't have one, you could be fined. (A broken machine is the exception = "non valido") BITs are only 1,50€ each. Maybe buy a couple to have on hand. Bus tickets can be used up to 100 minutes. Metro is one-time use.
Rome is fun, beautiful, exciting, awesome and awe-inspiring. I hope you 'love' it.