You can get cheap food in Rome, the problem is that if you don't know the city well or are condemned to eat out in the centre, the prices go up.
In Campo de' Fiori you won't have any trouble finding local food shops and there's also a food market right in the square in the morning. Cooking at home is cheap. For lunch in Campo de' Fiori try the Forno di Campo de' Fiori, a traditional bread shop that makes one of the best kinds of pizza bread in Rome. With your back to the Giordano Bruno statue in the middle of the square, the Forno is at the bottom of the square to the left.
You buy the pizza and eat it walking around, since it's a bread shop with no sitting area. You can choose how much you want to eat, but you'll be spending about 2 euro each + about 1.50 for a can of soda or 1 euro for water.
At lunchtime if you go to a restaurant you'll be spending between 15 euro for a set menu (not always good, try to go where you see Italian office people having their lunch as your best bet) and 20 euro. For this you should get a pasta dish and a second course (meat) ending with coffee. Obviously if you have special wine the price will increase.
At night you can spend about 20 euro at a pizza restaurant or if you only have one main course. Again if you have special wine it will cost more.
In any case, for a sit down lunch or dinner in a restaurant 15 euro is cheap and 20 is more normal and then up from there.
Chinese restaurants are still quite cheap but other ethnic restaurants (Thai, Indian) are generally expensive (think 30 euro upwards per person). Sushi is very expensive (40 euro minimum per person).