Restaurants in Rome
There are a great many wonderful restaurants in Rome, and a great many terrible ones: as in other areas, a huge industry has grown up around selling low-quality to tourists at high prices, and this has gone some way towards marginalising the best of Rome's restaurants. (Why work hard to prepare great food cheaply for locals when you can defrost a slab of lasagne for a tourist and name your price?) With a little research, though, you can make your dining experiences memorable for the right reasons.
One of the ways that tourists get into trouble is by tiring themselves out on a long morning spent sightseeing, and then sitting down at the first place they find; sadly, the 'first place you find' when you're on a tourist route is rarely a plce worth eating in. Good restaurants are rarely on the main tourist routes, and never have waiters outside harassing passers-by.
The best approach is to plan ahead, know where you're likely to be around lunchtime or dinnertime (Romans tend to eat around 8pm, or later), and select a couple of choices. If you have an iPhone, there are an increasing number of apps that help you sort the good from the bad: one is Rome for Foodies by Parla Food.
Finding value for money
Eating out is part of life in Rome. It used to be said that it was cheaper here to eat out than to purchase the same ingredients in a supermarket and cook them at home. That's no longer true, but you can still find cheap places to eat if you know where to look.
Remember that if you eat in the tourist centre, you'll likely be paying over the odds and the food may not be the best. There are exceptions to this, however, and you'll find a few recommendations, along with tips on tipping etiquette, in the page on restaurants in Rome.
Smoking is banned from all enclosed public spaces in Italy, and restaurants are no exception. Many restaurants have either a courtyard, or tables on the pavement where you can enjoy a cigarette between courses, but if you're a smoker you may want to check this before committing yourself; or just pop out for a smoke between courses. (Italians have been known to complain that, by forcing them outside for a smoke, the government is saving them from cancer but condemning them to pneumonia instead.)
Vegetarians in Rome
Although the more tourist-oriented restaurants in Rome have now cottoned on to vegetarianism, if you go out of the way you're still likely to find yourself confronted with a waiter who doesn't understand the concept. (Vegetarian? No problem: have some chicken!) Meat and fish are important parts of the Roman diet, and many dishes that might appear vegetarian will turn out to at least contain anchovies.
Be patient and explain what you want, and don't be afraid to ask what's in the different dishes. But also, be as flexible as you can be: if you're the type to periodically break over a bacon sandwich or a cheeseburger at home, you really ought to let yourself break when given the chance to experience Roman cooking at its fullest!