We went to Rome in May for the first time. Oh, god, I want to go back, I loved it!
We stayed in a hotel right at the Pantheon and walked miles each day so you'll need your comfiest shoes or trainers.
Taxi to and from the Ciampino airport is easy. The city taxis are all one price from there to anywhere within the city walls so you don't get ripped off. Think it was around 32 euro, can't remember.
Go to the Trevi fountain earlier than you think. We gave up getting anywhere near it in the afternoons and went at 8am one morning instead. It was still fairly busy but at least we could get down to the front to get a photo.
Be prepared for crowds. And the cruise crowds are the worst, especially in the narrow back streets. So slow and shuffling. The crowds in the Vatican museum were something else, even though we'd got early entry.
Be prepared to be totally wowed by all the historical sites. I mean, they are EVERYWHERE. In the taxi from the airport I just kept shouting "oh my god, look at that!" every 5 mins. The taxi drivers must just roll their eyes....
Ask your hotel receptionist for recommendations for particular types of restaurants. Ours recommended one for pizza in a side street just near the Pantheon. It was so good we went back the next day. Leave room for gelato though - I ate too much each lunch and didn't end up having any gelato because I was too full!
You will be surrounded by stylish Romans. So while you want to be comfy, you don't want to look scruffy. I made the mistake of wearing jersey jogging bottoms and a loose (saggy!) top during the day plus trainers, just for comfort. Oh god, I really stood out as a tourist and felt uncomfortably scruffy so next time I go I'm going to make sure I look a bit smarter.
Only have a cappuccino at breakfast. They don't tend to drink them at other times and you know, when in Rome......
We visited:
the Pantheon, the Colisseum and Forum, the Vatican museum, Trevi fountain, Spanish steps, Trastevere for an evening meal, the campo di fiore, Piazza Navona, Piazza del Popolo and Villa Borgehese gardens. Largo di Torre Argentina was not accessible when we went but you could walk around the edge. Mauseleo di Augusto was closed too but was good to see from the outside anyway.
I recommend you use Google maps in advance to plan what you want to see because you want to know how far between the sights and which are in the same area so you're not walking backwards and forwards between them.
Just walk, walk, walk, soak up the atmosphere, explore the neighbourhoods, stop in the bars, on a piazza for a coffee people watching. You'll want to take LOADS of photos.