Rome by a week
First about food, try to find places where locals eat, plan them ahead, cause most of the places, near tourists attractions are worthless and expensive, if its possible, get tickets on the museums and galleries ahead online, it saves hours. Try to use metro, instead of a bus, Rome has insane traffic delays, to go for the airport , use train, fastest, stable and defenately worth it, bus its a gamble and private cabs, well, gamble, taxi is not preferable, since again, lot of traffic.
Insider Tip:From personal expirience, get an offline maps for your smartphone gps, paper map might be tricky for unexpirienced travelers. Borghese gallery needs to get an appoiment in advance, so try to get a ticket online few weeks before, same goes to the museum of Vatican. Try to get some audioguides for your phone, or if you visit museums and galleries, which most of them arent national, try to get an audioguide, cause most of the times i've stuck watching whole museum without single english tab on the wall.