If you plan ahead, eating out in Rome will be one of the highlights of your trip. A memorable meal needn't be expensive either: eating out is part of life in Rome. It used to be said that it was cheaper to eat out in Rome than to purchase the same ingredients in a supermarket and cook them at home. That's no longer true, but you can still find good value in restaurants if you know where to look.
Finding a good restaurant
There are a great many wonderful restaurants in Rome, but there are also a great many terrible ones: a huge industry has grown up around selling low-quality food to tourists, often at high prices. With a little research, though, you can easily make your dining experience memorable for the right reasons.
One of the ways tourists get into trouble is by tiring themselves out with a long morning or afternoon of 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 restaurant worth eating in. Good restaurants are rarely on the main tourist routes (although they might be close by), and they never have waiters outside harassing passers-by. It's usually the other way round: a good trattoria or pizzeria will often have a group of would-be diners waiting for a table.
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 even later, but restaurants are usually open earlier than that), and choose a couple of options. If you have an iPhone, there are an increasing number of apps that help you sort the good from the bad: one is Katie Parla's Rome.
Smoking during your meal
Smoking is banned from all enclosed public spaces in Italy, and restaurants are no exception. Many restaurants have outdoor seating of some sort, 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 confirm that smoking is permitted (or look for an ashtray) before sitting down; alternatively, 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, perhaps an exaggeration given the mild climate.)
Special dietary needs
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 might 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 comfortably be: if you're the type to periodically crack over a bacon sandwich or a cheeseburger at home, you really ought to let yourself crack when given the chance to experience Roman cooking at its fullest!
Gluten-free options are becoming easier to find in Rome. As is often the case, people choosing to avoid gluten for lifestyle reasons have made gluten-free food more commercially viable, and also increased the availability of gluten-free food for coeliacs and others that need it. There are some dedicated gluten-free restaurants and bakeries in town, and many other restaurants will happily prepare something for you. If cross-contamination isn't an issue for you, it might be worth picking up a bag of gluten-free pasta before you go out for your meal, and asking the waiter of they can use it to prepare your chosen dish. Gluten-free pasta, bread, and snacks are available in most large pharmacies, with a more limited range in larger supermarkets.