The traditional home of antique shops in Rome is on Via dei Coronari and Via Margutta. (If you fancy combining the old with the new, there are also some interesting modern art galleries in this area.) One of the best known places is Veneziani on Via Margutta.
To improve your chances of getting the genuine article, it's wise to stick to the stores and avoid street sellers, and make sure that you get a certificate of authenticity.
One alternative if you're looking to buy something antique without spending a fortune is antique prints. A couple of good places for this is the antique book and print store on Largo Febo and the one on Via dell'Umiltà.