A (pescatarian) freind went last year, I wrote this for them, may help
Things to see:
Pantheon (free) Largest unreinforced dome in the wold and it's 2000years old!
Pantheon · Piazza della Rotonda, 00186 Roma RM, Italy
St. Peter's Basilica St. Peter's Basilica · Piazza San Pietro, 00120 Città del Vaticano, Vatican City, free to look in the church, you can pay to go up the dome, if a nice day its a great experience. The pope gives his speeches in the piazza outside, time it right and you may see him.
Trevi Fountain (free) If you are out at night, worth another look, get lit up and you have it to yourself.
Basilica di Santa Maria in Ara coeli (You will probably walk past it without noticing if you go to the colloseum) (free)
Basilica di Santa Maria in Ara coeli · Scala dell'Arce Capitolina, 12, 00186 Roma RM, Italy
If you see a church and you have time, just go in, you will be amazed at some of them.
If you are passing:-
Trastevere An area of rome on the vatican side of the river, not done much there myself, but should be handy for you, more of a "locals" area supposed to be very nice.
Campo de' Fiori Campo de' Fiori · Campo de' Fiori, 00186 Roma RM, Italy If you are passing, there is often a proper italian food market on, which is interesting.
Largo di Torre Argentina Largo di Torre Argentina · Largo di Torre Argentina, 00186 Roma RM, Italy. more ruins, now home to abandond cats. But it is where Julias Ceasar was murdered.
Snacks
Giolitti Giolitti · Via degli Uffici del Vicario, 40, 00186 Roma RM, Italy Best ice cream (gelato) I found in rome! Also sit outside, have an Aperol Spritz. and watch the world go by.
MAMI MAMIPIZZA - Google Search
Slices of pizza, they cut off what you want, weigh it and heat it. Also do cones of fried fish/arancini/vegetables.
Pasticceria Boccione Pasticceria Boccione · Via del Portico d'Ottavia, 1, 00186 Roma RM, Italy
In the Jewish Ghetto if you are passing, buy a slice of cake/tart. Looks a dump! and run by famously grumpy old ladies.
Coffee
Don't order an "Americano"!!! "Filter Coffe/French press" don't even exist out there!
have an espresso, cappuccino or macchiato. If you see someone wearing a uniform going in or stood at the counter, it will be good.
Eve Meal
Be warned, in the evening, the good restaraunts dont open until at least 730pm! If they are open before they are probably touristy. Have a look at what you fancy lucnch time.
If you stop for an alcoholic drink, you always get a snak included to go with it. sometimes its just crisps, sometimes its almost a meal in its own right.
Local vegetarian dishes to look for
Tonnarelli Cacio e Pepe - Pasta with cheese and pepper
Roman artichokes (the romans love artichokes)
Frito Misto - breaded mixed fried fish.
Anti Pasti. Normally lots of Salami! but often you can get Cheese, vegetable of fish.
Limoncello - Lemon liqueur, try one after a meal.
Restaurants i've been to that were good:
Osteria Pasquino Osteria Pasquino · Piazza di Pasquino, 1, 00186 Roma RM, Italy Good pizza, and the best Carbonara I had in rome. (not that you can eat it!)
Virginiae Ristorante Virginiae - Cucina Tipica Romana · Via di Parione, 41, 00186 Roma RM, Italy (prob best book) If you want to treat the lady! not ridiculously expensive for the setting.
Ristochicco ristochicco - Google Search if you have the foccacia, share it, its huge!
I Pizzicaroli i-pizzicaroli-piazza-navona - Google Search I really liked this place, Anti Pasti Platters of local produce, may not be much vegetarian wise though! Will be cheese ones, though.