Melissa,
As the others have mentioned, there are a variety of ways to dine inexpensively in Rome. Buying a sandwich (or whatever) and a bottle of juice or water at a grocery store (Alimentari) is one solution.
Many Hotels in Rome provide a "free" breakfast. These tend to vary a bit between properties, but will be more of a "continental" style rather than what you're probably used to at home. Of course these aren't really "free", the cost is built into the room charge.
Dinners are probably the biggest expense for most travellers. The typical Italian meal consists of three courses - Antipasti, Primi and Secondi. If you get the "full meal deal", it gets expensive really fast (especially for two people who may also order a bottle of wine). You can moderate this to some extent by sharing - both order the antipasti, one orders the primi and one orders the secondi. If you're in the habit of ordering a "starter" salad, note that there are usually only two choices for dressings - oil & vinegar or vinegar & oil! Eliminating dessert is another way to save money.
Rick's Guidebooks always have listings for a selection of restaurants in various price ranges. It's a good idea to pack it along in your Daypack so that it's available for reference.
Regarding touring, DON'T rent a Car!!! Rome is one of the worst places to drive, and it's a good idea to exercise caution even as a pedestrian. Red lights and other traffic signals are often seen as more of a "suggestion" than a law by drivers in that part of the world (it gets worse as you head south into Naples!). The Metro, Buses and Trams are more than adequate to get you around Rome. Again, details in the Guidebooks.
Be sure to wear your Money Belt and watch your valuables, especially on the Metro and of course the #64 Bus which has a bit of a "reputation".
Happy travels!