The San Pietro station is a local station within walking distance from the Vatican. From the Termini station, which is the main station in Rome, you can take a regional train to the San Pietro stop, which would take about 20 minutes. You would pay to board this train locally.
The Colosseum in Rome is only about 2 Metro stops from the Termini station. There are even stunning views of the Colosseum from on board the Metro as you approach. I believe the stop is even named the 'Colosseum' stop or something similar. You would also pay to board the Metro locally.