Waterman’s list is good.
There are some problems. St. Peter’s is off to one side of the Centro storico, and the distance, although it may not seem great, is considerable, especially is one is tired after a day at a conference. This is compounded by Rome’s impossible traffic at 6 pm, and the difficulty in just hailing a taxi. About two years ago I had to walk from Castel Sant’Angelo to Termini (the station). It’s about two miles, but in the pouring rain it seemed more like 10.
Possibly the best thing is to try to pick up a taxi at the top of the Via della Conciliazione - the avenue that leads from St. Angel to St. Pter’s Square (Near the great colonade) and go to Piazza Venezia where the Vittoriano (Wedding Cake) is. The Capitol/Campidoglio is behind and to one side of the big white monument. Beautiful square with replica of equestrian statue of Marcus Aurelius. Fine architectural piece the whole layout, incuding the Cordonata (a sloping staircase with statues of the Dioscuri, Castor & Pollux, at the top.)
Behind the central building (the one with the Fountain) is the Roman Forum. Taking the ancient Scalae Gemonianae (on lhs) you pass down by the Tullianum/Mamertine prison and passing the the Forum of Julius Caesar, you arrive on the modern Via dei Fori Imperiali. This leads past the Forum of Trajan (lhs) and the original Forum (rhs), with views up to the Palatine (Palace) Hill. The Viale (avenue) led directly to the Colosseum. (Be careful of bags, cameras.) Just beside the Colosseum is the arch of Constantine. This is probably enough for one evening. You need to eat.
Another itinerary:
Taxi to Piazza Navona - although it might be possibile to walk there… The piazza is what remains of the Stadium of Domitian, and it still retains its shape. From here you are into the narrow streets of the Campo Marzio. Take care of valuables and keep them inside outer garments. It’s a short and interesting walk to the Pantheon. This building is worth seeing both inside and outside. Hope it’s open in the evening as Waterman suggests. It didn’t used to be in my days in Rome. From the Pantheon it’s not too far to Largo Argentina where there are the remains of some very old temples that pre-date the Empire. From here the Campo dei Fiori with its statue of a brooding Giordano Bruno is within reach. You are now in the Jewish quarter of Rome. After this, time to eat again.
I could go on, but it really is up to you to get a decent map and do at least a little reading up in the few days before you leave.
As to eating out, Rome is full of restaurants, trattorie, and ostarie - in descending order. Where you eat will have an important bearing on the price. If you see a place with lots of locals installed there, then it’s a good chance that it is reasonable and the food is good. If it’s packed with non-Italians, chances are it’s expensive and the food is just ok - and possibly rude service. Also places in view of prime sites will attract higher prices.
In case you think I am not including a lot, the possibilities for mental indigestion in Rome are endless. If you overdo things, memories get garbled. So it’s probably a good idea to keep a notebook of where you go.
I am an old hand at Rome, so I am stressing the ‘dangers’ of pickpocketing and bag snatching from motor two-wheelers. I have seen it many times, and I even have had my shirt pocket picked by Chinese. Yes, Chinese, working in tandem on the 64 bus. That’s the one that goes from Termini to St. Peter’s and it’s renowned for this. So keep your wits about you, and have a sense of purpose, always remembering that anywhere is a likely place for an unfortunate event. Don’t be frightened or put off. It happens in every city. Just keep alert and don’t do silly things like taking out purses or cash in public places.
Check out what temperatures are like. It’s about 15 C there at present (multiply by 9, divide by 5 and add 32 = F) = 59 F during the day. The evenings will be a lot cooler, if not colder.
Have a great time.