You can visit a lot of the sights in walking distance from the main shopping street, Via del Corso including - Trevi fountain, the Spanish Steps, Piazza del Popolo, Piazza Navona and the Colessuem which is a bit further on. Visit the Parthenon as well and have gelato or cannoli in a place nearby… More
La Prosciuterria Trevi on Via della Panetteria is a cool place. Small and intimate but fab food. You order at the counter before sitting down but a very authentic experience.
There's an area clise to termini called Monti where there are a lot of cool places and shops. It's full of artists, artisans, vintage shops, bars and restaurants. You should definitely check it out!