On your arrival day, you should do something that is on you bucket list to get it out of the way since you don't have a full day. My personal must do is a Private Pizza & Pasta class! I always wanted to make pizza and pasta in Italy. I recommend you utilize Rome Pizza School if you decided to attend a class.
For Day 2 and Day 3 you need to make sure you are ready for full days! You should be out and about around 7:00am and your day could end about 8:00pm.
-Spanish Steps/Trevi Fountain/ Pantheon- I would do these first if you want great photos (no crowds and they are all walking distance, especially if you stay in a hotel nearby)
-Roman Forum and Palatine Hill/Circus Maximum
-Colosseum (If you have time, stop by Trajan's Market which is the oldest mall in the world)
-Pantheon (If you would like a tour)
-Via Borgognona, Via Condotti and Via del Corso
-Musum and Crypt of Capuchins
*I recommend having lunch either in Piazza della Rotonda (by the Pantheon) or in Piazza Navona
-St. Peters Basilica
-St. Peters Basilica Dome
-Sistine Chapel and Vatican City Museums
-Campo di Fiori
-Ponte Sisto/Basilica di Santa Maria (in Travestere)
-Janiculum Hill AKA The Gianicolo Hill (in Travestere)
-Tiber River Cruise
*I recommend having lunch around Campo di Fiori or in Travestere
Please be aware that these places are extremely crowded and you can stand in line for hours. I recommend that you pre-purchase your tickets or get a tour that has all of these attractions. Remember I am a Travel Agent and would love to help you with your upcoming trip!
"When in Rome, Do as Romans do"
P.S The Trevi Fountain is one of the oldest water sources in Rome. Make sure to make your wish! Throw your coin with your right hand over your left should to ensure you will come back to Rome sometime in the future.