Arvind- I believe you are arriving in Italy from other places farther north. Are you planning on sleeping in a different place each of the 3 nights, or just passing thru Pisa on your way to Rome? You may want to visit Pisa first. The famous Tower is located together with the Duomo and the Baptistry, so you could catch 3 sights in one place. The Cinque Terre, 5 special towns along the coast not very far north of Pisa are worth visiting on another trip, but you won't have time this trip. Hit Pisa (maybe Lucca, too) then get to Rome before the end of the day.
Rick Steves has a Rome guidebook that also includes Naples, but not Pisa. Or you can get his full Italy guidebook, which is more than you need for this fast trip. The Rome book also includes information on sights within several museums and walks in neighborhoods within the city, so you can do your own museum and city tours. Note that not every sight is necessarily open every day of the week. Depending on what you see, purchasing a Roma Pass for each of you could save time and/or money.
We visited Rome for a whole week in 2012 and only saw part of it. Perhaps hiring a local tour guide to show you the highlights might make good use of your time. At The Vatican, you can see St. Peter's square outside, and do a quick visit inside St. Peter's Basilica, a really big church that will take some time to go thru. Michealangelo's Pieta is just inside the doors on the right, but you may want to venture further inside to view more. Climbing the dome (with an elevator part of the way) will take more time and energy, plus a fee, if you decided to do that. The Vatican Museum, including the famous Sistine Chapel with Michaelangelo's ceiling frescoes, is huge and will take some time to walk thru, even if you don't stop to look at much of the art, so you might decide to skip the museum, although it is outstanding.
The Borghese Gallery requires a reservation, only allows you 2 hours to visit inside, but it's an amazing building full of fabulous art.
For the Colosseum, purchase the tour that gets you into the lower level. See the Pantheon. We also liked the National Museum of Rome, the Capitoline Museums, the Palatine Hill, and the Roman Forum, but each can take some time to view. These probably aren't on everyone's Top 10 list of sights in Rome but we also really liked the Villa Farnesina and the San Clemente church with its 12th Century street level, 4th Century lower level, and 2nd Century subterranean level.
If you stay in Rome each night, you can even visit Naples (the original home of pizza) on a long day trip. Rick suggests a blitz visit that involves leaving Rome by train early in the morning, visiting Pompeii, which was buried when Vesuvius erupted almost 2000 years ago, seeing the Archaeological Museum in Naples (which has most of the art recovered from Pompeii), seeing some of the city of Naples, then retuning to Rome late at night. You can see Mount Vesuvius looming over Naples from a distance, but going on a tour to drive or hike up the mountain will take time and energy. Another very interesting ancient Roman city buried by the Vesuvius eruption is Herculaneum, which is smaller than Pompeii and closer to Naples, so Herculaneum might let you see an archaeological sight but take less travel time, and with less walking, than Pompeii.
We stayed at the Rome Airport Hilton our last night, which made it really easy to catch our early morning flight home. Their incredible breakfast buffet had foods from all over the world, something for everyone, and the room price was surprisingly good in December 2012.
