I'll give you practical advice here, because any travel guide you buy will tell you what the nice things are to go see. I bold things I think are super important.
Definitely learn to use the bus system the minute you get there. The metro there is ok, but it doesn't go a whole lot of places. Taxis may rip you off.
Bring a water bottle to fill up. There are tons of free water fountains that dispense water from the aquifers. They are not your typical water fountains.
My recommendation, Capitoline Museums:
A lot of the cool statues you associate with Rome are housed here. It was my favorite of all the museums we visited.
Vatican:
Check the Vatican schedule ahead of time. If the Pope is doing something outside, like giving a speech or mass or whatever, that's a good time to go visit the museum (cuz everyone wants to see the pope not the museum). Otherwise it's packed. Get ready for a long wait or get there super early in the morning. See the sisteen chapel, be quiet inside, bring a camera you can take pictures without flash.
St. Peter's Basilica:
If you go upstairs to visit the top of St. Peter's, pay to take the elevator for the love of all things holy. Go upstairs though, you don't fully comprehend the size or grandeur of it until you're on top.
Trevi Fountain:
Avoid the peddlers.
Have the gelato. Return for more gelato.
Roman Forum (Foro Romano):
Bring sunscreen and water, there's not a lot of shade and it's hot in Rome in the summer.
Colosseum:
Avoid the picture takers and the pick pockets who inevitably linger here.
Circus Maximus:
I was honestly disappointed. It's a big grass field now, nothing was going on there when we were there.
Galleria Borghese:
Call ahead and make reservations the minute you get there or now. They book in advance and only a limited number of guests per day get in. Walk around the grounds afterward. I spent a full day in the surrounding grounds, there are some great people watching places. There are also some great photo ops on the hills overlooking the city. Bring a football

Spanish Steps (Piazza di Spagna):
Crowded. Avoid the peddlers. There's an amazing vegetarian restaurant somewhere downstairs to the right if you're coming down the steps, forget the name. It's in a side street. Food was to die for. Shops around the steps are expensive.
Drink espresso. Eat a lot.
Have wine with everything.