What is the name of the district that is considered to be the city center of Rome?

Transportation and getting around wise, your best bet would be to stay somewhere near the Termini train station. That is where the long haul trains arrive, and you are pretty much 10-20 minutes away from pretty much every major attraction. Being close to the Termini station can be helpful if you plan to go/come from Florence, or Civitavecchia, where the trains arrive at around the mid night, and there would be no metro to use at that time.

I stayed in a couple hostels during my stay, and I really liked one near the Vittorio Emanuele metro station. I'm no way affiliated with them, but here is a link if you'd like to try it out. The staff was very friendly, but note that the surroundings were some Indian restaurants and one would say it's a dodgy neighborhood. I didn't care about that.

For a different environment, try to book somewhere in the north western region, where the Vatican is located at. There is no metro line too close (the closest is Ottaviano), but I never felt any tired walking from a nearby metro station.

There are trams and buses that connect an otherwise star-like metro system. It's not the smoothest way to travel (there were two stations out of service during my visit, although they provided alternative transport via a bus). The tourist pass wasn't that worth it, but 1, 2 or 3 day transport cards were well worth it.

You can read a lot more about there from this nice page. Have fun, and be safe!