The Vatican is situated to the east of Rome's city center and can easily be reached using any of the available public transports in the city.
The closest station to the Vatican is the Ottaviano-S. Pietro station. Line A of the Rome Metro stops at this station every few minutes. Both the Vatican Museums and St. Peter’s Basilica are a 5-minute walk from this metro station.
The Vatican also has a train station, St Pietro, and serves commuter style trains travelling between Rome and Termini Stations. However, it is a 10 minute walk to St Peter's Square from the station.
If you're planning to take the bus to the Vatican, then you have two options that stop on either side of the Basilica.
Buses 32, 81, 982 stop at the Piazza del Risorgimento (left of the basilica) which is a 10 minute walk from the Basilica.
Bus No. 64 stops at Via di Porta Cavalleggeri and lies to the right side of St.Peter's Basilica. This is slightly closer than the stop at Piazza del Risorgimento.