There are many cafes and restaurants in central Rome - almost on every street ranging from cheap to expensive.
You can search and reserve a table at several of Rome's restaurants at:
Restaurants are generally open between 12:00 - 14:30 and 19:00 - 23:00, with cafes open all day selling pizzas and sandwiches etc
There are a few vegetarian cafes and restaurants in Rome.
See www.happycow.net for a list. Although, you can normally find something vegetarian to eat in most cafes and restaurants in Rome.
Some useful phrases...
I'm a vegetarian: Sono un vegetariano.
I'm a vegan: Sono un vegetariano integrale.
(Note: Many Italians will think you eat fish even if you tell them you're a vegetarian so it is best to use the following)
I don't eat meat: Non mangio le carne.
I don't eat fish: Non mangio le pesce.
I don't eat eggs or dairy: Non mangio le uove e non bevo latte