Friends of ours recently traveled to Italy for the first time and asked us for some tips for Rome, as we'd just been there last September. They had a great time and loved our suggestions, so I thought I'd share them here also..
I would definitely recommend doing an organised tour for this, as this means that you can skip the very long lineup going into the Colosseum. I would recommend hats, sunscreen and definitely sleeves - a lot of it the Colosseum is open air, and I got burnt when we went. This is the tour we did, it was absolutely amazing, the tour guide was fantastic! Very in-depth tour, and after the Colosseum the tour continues onto Palatine Hill and into the Forum Romanum.
I would definitely do the morning tour, so you can do lunch straight after, and then do some other stuff on your own in the afternoon.
We did another tour for this, again, you can skip the lines and get great English commentary (as for the tour above). Another very knowledgeable guide, and i think you do need someone to guide through the Vatican museum and explain the background to all that stuff, i think it would be overwhelming to walk through by yourselves. Also goes into the Sistine Chapel and loads of explanation about the background to the frescoes. Link to the tour.
Remember to wear shirts with sleeves and pants/skirts below the knees.
All the other sights we just did by ourselves. Your hotel should be able to provide you with a map detailing where all the sights are. We stayed about a 15 minute walk from the Colosseum, so we walked everywhere, except the Vatican, as that's on the other side of the Tiber. But a lot of the other stuff was within walking distance.
Spanish Step, Trevi Fountain, Pantheon can all be done in a day.
Take your time for the Peters Dome. there can be very long lines. Again, sleeves covering shoulders,and pants/skirts over the knee, they will turn you back if they find your outfits too revealing. This is an amazing structure, make sure you have heaps of time here; there are loads of artworks, amazing architecture. You can also go into the catacombs below the dome, where all the popes are buried (and the faithful come and pray there, too.) An additional thing you absolutely HAVE to do at the dome is climb it. It's an additional cost, but so worth it. (not that expensive, actually) You can go in the actual dome-bit at the top. First you can look into the Church from up there, and then climb up further and go outside the dome. Most amazing view of Rome! Definitely do this, it's so worth the climb. It's a bit tight up there, but unless you guys have major claustrophobia, you should be fine.
If you're looking for somewhere to eat, there are great pizzerias in Trastevere, which is an old Suburb in Rome. Apparently all the locals go there. There are stacks of restaurants there, you should be fine anywhere there.
If you're serious foodies, there is a restaurant that serves cuisine that most people probably haven't eaten before - Roman Jewish food. Our Colosseum tour guide put us onto this place, and it was amazing. The lineup was massive when we were there, but it was on a Sunday lunch, so that was probably why. Might be good to book. It's called Ba'Ghetto, in a lovely old Roman neighbourhood. Via del Portico d'Ottavia, 57, 00186 Roma. Visit the Ba'Ghetto website.