Most people go to Italy to enjoy the food, and rightly so, since Italy has the best food in the world!
But there are few things to keep in mind about the food in Italy.
The first thing is that food in Italy comes at meal time and not much in between.
It seems like a strange thing to say, but in the US people are used to having food 24 hours a day, Pizza at 3 am is not unusual, and most restaurants are open from lunch time to closing time. Things are changing a little, with some places open later or all day, but for the most part, well, in Italy food places have schedules.
Breakfast is usually something at a bar (Not a booze bar, a cafe` bar) like a pastry and coffee or cappuccino or some other warm drink and it's usually early in the morning.
Lunch is usually from noon to 2 or 2:30, and restaurants don't open until lunch time. After lunch, most of them close and there is no food. Usually around 2 the food is gone and restaurants don't serve you anymore.
Dinner is usually late in the afternoon, with most places opening after 6 or 7. Dinners in Italy are a late affair. 8 or 9 pm are common times for dinners. Restaurants are full of people eating dinner between 9 and 10 pm.
So, don't expect to find a restaurant serving dinner at 4 pm. Unless it's a fast food place or a pizza al taglio place. But then, the food that you will find is leftover from lunch, shriveled up and cold.
So the main advice is eat at meal times, have breakfast early morning, have lunch around 12 or 1 pm and eat dinner after 7. Otherwise you won't eat at all. For people's convenience the stores follow schedules too, so stores open in the morning and close at lunch time, and open again in the afternoon and close at dinner time. Museums for the most part have full day schedules, but the smaller ones might close at lunch too. Food places are open in the morning, and close at 1 or 1:30. Most are closed on monday... Churches have schedules too, they also close at lunch time but you also need to be aware of when they have Mass. You don't want to meander in a church to look at art while Mass is going on.
So, in any case, check the schedule.
As far as food itself, one thing I would say is avoid the touristy places, the kind near tourist attractions, with the waiters outside trying to get people in their place. Their food is usually an Americanized version of Italian food. You'll be better off eating at places where the locals eat.
And don't expect the food to be like what you find in the States, because American Italian food is different from Italian food. But it's better!
If you are vegan, make sure you ask a lot of questions about the food. Be aware that in some cuisines lard is commonly used as a fat in breads and pastry.
Make sure to specify what you eat and what you don't. Some people think that fish and chicken are not meat, or that butter is not dairy. It helps to learn few words for what you eat or do not eat. Or make a card with a short list. The vegan diet is a little more common these days, but still, if some foods are an issue (allergy or unclean etc) it's better to be safe than sorry.
I will be listing places to eat along the routes I plan to put together, I will be suggesting vegan places, obviously, because those are the only places I know.
One word about ice cream, most bars (Cafes, see above) have ice cream, but not all ice cream is the same. Look for artisan made ice cream. They usually look like this:
Some places have some vegan ice cream, some places have a lot of vegan ice cream, even vegan cones and whipped cream. Ask, you might be lucky and score a great gelato!
In most places you decide before hand what size you want and then go pay at the cash register, get a ticket and take it to the ice cream counter and tell the waiter what you want, cone or cup, flavors and whipped cream or not. Some places you get the ice cream first and then pay. See what the locals do.
Here are some links to articles about gelato in Italy and the best places to go to.
This is a list of best places in Rome, and I personally recommend the Gelateria della Palma, near the Pantheon.
I have gone there several times and they have a large selection of vegan ice cream and cones! Soy and rice based ice cream. It's very good!
Another outstanding and old ice cream place is Giolitti, not far from the Pantheon also, and they do have some vegan flavors, mostly fruit ones, but they are not labeled as such. If you are not vegan, try their Zabaglione and pistacchio flavors, they are the best, as far as I remember from when I was a kid. If you are vegan, I had their fig and blueberry and few other vegan fruit flavors and they were outstanding. Ask questions to the servers, but don't expect too much chatting if they are busy. If you go when they are slower, they can tell you and even have you sample few flavors.
Considering all the walking you will do, you will burn the calories pretty easily and you will be able to have ice cream more than once a day! Enjoy it because you won't be able to have ice cream anymore afterwards in the States!