The SO and I will be in Rome for a week in late April. Stuff to do? Stuff not to do? Travel tips?
Do as the Romans do.
Sorry, couldn’t resist! 
Buy a Roma Pass at the Tourist Info Points. Lets you into many Museums and historical sites (not the Vatican ones) and the public transport. There is a Michelangelo statue of Moses in St. Pietro in Vincoli, it’s free.
Trevi Fountain is my favorite part of Rome. Day or night.
Have some Gelato. No, have LOTS of Gelato!
St. Peter’s and the Vatican are of course a given, as are the Pantheon, the Coliseum, and the Forum.
If you make it to the Via Veneto on the north side of town, check out the crypt of the Cappuccin Monks in the Church of Santa Marria della Immaculate Conception (or something like that) at 27 Via Veneto; thousands of bones of dead monks, artfully arrayed.
Cafe Barroco in Piazza Navonna is a great place for a snack or a beer and people-watching.
Also - if you order something at a sidewalk cafe, they will charge you twice as much if you want to sit and eat it there as they will if you get it to go.
Rome? Nah, not much to see there.
Very true, but the flip side of that is that if you get a table and buy an espresso and a bun the table is yours for as long as you want it, perfect for people watching.
I felt the best gelati was found at the Gelateria Della Palma, Via della Maddalena - just near the Pantheon, which is also well worth the visit. I also had some amazing food at Otello alla Concordia (apparently one of Fellini’s favourite restaurants as he lived nearby) which is in Via della Croce 81 near the Spanish steps.
Other than that - walk. Everywhere is lovely and fascinating and such fun to poke into small shops down little alleys etc. Make time to peoplewatch, particularly during the evening passeggiata. Don’t order a latte (it’s milk) and don’t order a cappuccino after breakfast. Remember when you sit down outside that the cost of the drink will skyrocket, so cheaper to have an afternoon macchiato at the bar as the Italians do. We also made sure to take advantage of the aperitivo - in the evenings bars will lay on free food to accompany your drink, sometimes as simple as a wheel of parmigiano and a knife to carve off chunks with, and sometimes enough food to make dinner unnecessary.
Yes, I travel to eat. I’m sure someone more cultured will be in shortly!
Oh, and be aware of people around you, particularly people creating a scene to distract you, or getting unusually close. Pickpocketing is not uncommon, but they look for an easy target. Pay attention, be aware of your bags etc and you should be fine - I’ve been there several times as a single female and never had any cause for concern.
I highly recommend the Colosseum underground/upper level tour. It costs more but you see so much more than the average person going into the Colosseum and, since you buy ahead of time, you skip the line. Normal tours let you skip the line, too, but you would only get to see the main floors. I got my tickets from this site, though I hear it’s cheaper if you call the box office yourself. Regardless, I suggest you call ahead to make sure that the tour is open since it has been closed off and on due to flooding the past couple of months.
The real plus side of this tour is that you get to see everything that you’d see on a normal tour PLUS getting to go into underground tunnels and under the partial stage, plus you get to go to the highest point in the entire building. The only people in these areas will be the people on your tour - they can get up to 40 people, I was lucky that mine was only me and the girl I was with. It’s a guided tour so they tell you about everything you’re seeing and answer whatever questions you have. It takes about an hour or hour and a half all told and also includes entrance to the Forum within the next 24 hours. Taking this tour was probably the best decision I made about my Italy trip.
Consider a day-trip out of the city. we used the Rick Steves guides with lots of ideas. We took a local train and bus to Tivoli and Hadrian’s grounds. Other options he has are Pompeii or the port of Rome but I can’t vouch for those trips. Or you could take an overnight (or two) day trip to Florence and/or Pisa.
Do lots of walking and take the subway, just watch your belongings.
I’d second this. I was in Rome with Bear Nenno and his missus at new year 2010/11 and the best part was definitely the Colosseum. I would suggest you lower your expectations when you come to food though. I was thoroughly underwhelmed by Italy’s cooking in general, considering how much good Italian cooking there is in London where I live. Compare that to France where pretty much anywhere that isn’t a roadside service station serves amazing food.