I'm going to Rome again at the beginning of Sept, which will be my 5th time.
I usually take my 16-85mm and my 70-300mm. I once borrowed a 10-20mm for a visit. The 16-85mm stays on the camera 95% of the time, and I've found it the perfect lens and range for city holidays. The 10-20mm would be useful occasionally, but not enough for me to purchase one. Hardly used the 70-300mm at all.
I have my Polariser permanently attached to my camera because I like the effect.
It only comes off for indoor and night shots. And speaking of night shots, I only take the tripod out at night. And yes, I take a tripod every time. There are a few pics of Rome in my 500px link below.
If you are going to St Peter's Basilica, get there early. Staying close to the Vatican will make that easier. The last two visits I've gone to the top of the Dome on the Basilica, I got there just after 9am, and went straight to the top of the Dome. It is a trek up to the top though, and it all get a bit 'funhouse' as the wall of the Dome curve on the way up.
By around 10am onwards the queues to get into the Basilica can stretch around the whole 'square'.
The Pantheon is a must visit, and there a wide angle lens would be useful.
It's my favourite place, and gets visited a number of times while I'm there hoping to get the perfect pic. I haven't got it yet.
As has been mentioned, a ticket for the Forum will also get you into the Colosseum, and I think the ticket lasts for two days, but check on that.
The Trevi Fountain is very good, though very busy. It's location is very surprising, and takes a bit of tracking down. Piazza Navona and Piazza del Popolo a nice open places, though totally different to each other.
Last year I went on a Segway tour on the last day, but I should have done it on the first. Pricey, but I saw bits of the city that I hadn't seen before even in previous visits, and places where I'm going back to next time to take some pics. I always wanted to have a go on a Segway too, and it was great.
Very easy up and down the hills on one of those, though the old centre is not very hilly at all. Tried to book a Segway tour before I went with various companies, and one company were willing to charge me double because I was on my own.
Very kind offer, but I declined.
If you fancy it let me know, and I'll search out which company I used. They were very good.
I always get a 7 day travel pass while I'm away, yes you can walk to most of it, but on a hot day being on the wrong side of the city, jumping on a Bus/Train is very nice.