A short list of things to do When In Rome... 

Check out The Colosseum, of course! Be sure to walk all the way around and enjoy the views.
Maybe go inside. Even though there are colosseums all over Europe, this is the most famous one.

Explore The Roman Forum. It's pretty cool to see the ancient foundations of this magnificent city.
Walk a real Roman Road. Realize you really did not do enough research on this area and
plan to take the tour next time you come back. Try to recall all you can about Roman Mythology.

Find something extra tall to climb and enjoy the view!

Marvel at the once far reaching Roman Empire. Be amazed at how much it has diminished.

Make your way across town to Vatican City if you're Catholic, interested in Religion and/or Christianity, or an art buff.
Be slightly annoyed that no matter how long you stand there waiting for people to move to get your
"perfect shot," someone will always step into your picture. Give up and move on.
Be sure to arrive early enough to go to the roof if you'd like.

Steal some Holy Water from St. Peter's.
Random thought: Does it lose it's holiness if it's stolen?

Later, check out the Vatican Museums and the Sistine Chapel, but don't take a picture of it because it's forbidden and they will confiscate your camera and/or SD card if you try. (I won't tell you how I know.)

Don't forget to see the Trevi Fountain. It's pretty spectacular and one of my favorite spots in the city.
Toss a coin in to ensure you'll visit Rome again!

Finally, go all the way back across town to see the Colosseum at night.
It's worth the trip no matter how tired you are.