Scooping....Rome is wonderful and everybody should go there, at least once, but it can be expensive !
I was there exactly 10 years ago next month, and stayed for 4 nights. We stayed in a small hotel in the Termini area, which was very good for transport links to most of the main attractions.
But it was only on our 3rd and 4th night that we discovered where we could eat very well and relatively cheaply.
The first two nights we ate in Piazza Narvona, and near the Spanish Steps. Wonderful food but mega expensive. Our Hotel concierge then told us about Trastevere !
https://en.wikipedia.org/wiki/Trastevere
This literally means beyond the Tiber and is located on the west bank of the Tiber. Its an area of higgly-piggly little streets, full of great value places to eat, all set around small piazzas. There are also very good value small Pensions and Hotels, and the area is about 25 mins walk from St Peters, and the Castle San Angelo.
Beware of the Italian "sitting down rule" though, in small bars and cafes. You will pay much more money to have a coffee or a drink sitting down, rather than standing at the counter !
One last tip...."latte" just means hot milk to Italians, not coffee as it does to us. We ordered two lattes and were brought 2 glasses of hot milk !
A phrase book will be worth its weight in gold.