Rome is a great city - my personal favourite by a country mile.
As everyone else has said, the obvious sites are the best ones to see! Most of them are within walking distance of each other as well. If you get the Metro Line A (the orange one) to Spagna, you can see The Spanish Steps. Just off the main square you have all your boutique shops and the main thoroughfare, Via Del Corso. If you make your way South East down Via Del Corso, you'll find the Trevi Fountain (a must-see) and then up the end of Via Del Corso, you'll find the Vittorio Emanuele monument which is really impressive. Behind this is the Coloseum and the Roman Forum which are also very impressive.
Then there are other attractions off the beaten path - the Pantheon being one of the most impressive. The Vatican is good - if you're stuck for time, skip the Vatican itself and just see the Vatican Museum which is really good.
For relaxing, there's Piazza Navona which is a big square with lots of coffee shops and restaurants as well as lovely fountains and monuments. If you're looking for a really special Italian meal, try
l'Orso 80 which is close by. I recommend the anti-pasta. Just don't order anything else as it's enormous!
As was already said - watch out for pickpockets, particularly in the Metro stations and on the trains. Be aware of who's around you when withdrawing money from cash machines as well.
Also, check the price of restaurants before entering. Romans have a great temperament but some of them are thieving cvnts, so be aware that they will try to rip you off. NEVER take a waiter's recommendation for a dish unless you know the price first. This is a typical Roman trick where you order the special and they charge you 50 Euro for it!