I stayed at La Griffe Luxury Hotel a couple of weeks ago...it's 10 mins walk from the main train station and was very good value (was part of an easjet holiday ~£400 each for 4 nights incl flights). You can walk to all the main attractions from there and it's just outside the immediate vicinity of the station (which has more than a bit of a run down feel). Only issue we had was that the room didn't have a window! Great rooftop terrace where you have your food and a bar as well (though a bit quiet)
If you don't mind a bit of walking, you won't need to bother with any public transport - Rome is very compact (population 2.8m)
We arrived on the Monday evening and left late of the Friday and did the lot at a very leisurely pace
There is a sort of triple ticket that covers the Capitoline hill, the Forum and the Colusseum. If the crowds are large at the Colusseum, go the the Capitoline hill first and buy the ticket there where the queue is minimal
If you go to the Vatican Museum, buy the tickets online before you go - believe me, the €4 per ticket 'booking fee' is money well spent...if you queue it's in the baking sun
Villa Borghese was worth a look, if you are into that sort of thing - just wondering around the park itself was pleasant
If you walk south on the Vatican side of the Tiber towards the river island there are loads of bars an food stalls down at the riverside, and quite a bit of live music a bit later on - wasn't mentioned in our guidebook