You really have only time for the major highlights so concentrate on those and dedicate as much free time as possible to the ancient historic center (this is not the Colleseum / Forum area but the area near the Pantheon / Trevi Fountain area)
This is where life is happening, a multiple block open air museum and is mostly pedestrian only. The other major sites are near main roads and are basically just busy with cars and tourists trying to avoid each other.
If you want to see the Vatican / St. Peters which most do that takes the good part of a day alone and is the one thing you really must book in advance so choose which day you will do that and try to keep that day otherwise clear.
You can order tickets online and they will be for a specific date and time. The time they are a little flexible on but the day they are not. This ticket purchased in advance will allow you to skip the line and saves hours of your valuable time.
Colleseum and the Roman Forum combine together well. Palatine Hill is definitely not as great as those 2 sites but still interesting. The real value for me of Palatine Hill is the line there is always short, so you can buy your ticket there and that ticket is good for all 3 places: Palatine Hill, Forum and Colleseum. The same ticket will have an hours wait at the Colleseum.
For this reason I would not pre buy my tickets to the Colleseum. Not sure if the underground tours need prebooking or not so might check into that if that was of interest.
Download Rick Steve's podcasts before you go and install on everyone's phone or ipod type device, he does some great free audio tours you can listen to inside the Vatican, St. Peter's, Colleseum, Roman Forum, Pantheon, etc...
They are all good and save you the need to buy any audio tours but the Roman Forum one is fantastic as there is virtually no signage there and the podcast tour brings the place to life.
Basic 3 day plan would be 1 day Vatican/St. Peter's, 1 day Palatine Hill, Colleseum, Roman Forum and 1 day ancient historic center. This leaves hours free each day to get lost and walk around hopefully in the historical center checking out many different piazzas there, enjoying the life of the city in this area, check out the different churches you walk past, venture across the river for dinner in Travestre 1 evening, have much cappuccino in the morning and gelato the rest of the day.
I would use your hotel area as a place to sleep mostly and spend my time in the historic center even if I was not staying there.