The Vatican is located off the 'Cipro' or 'Ottaviano' stops of Line A (the Orange Line) on the Rome Metro. From the Termini station (the main station in Rome), the Cipro stop would be 7 stops away on Line A and the Ottaviano stop would be 6 stops away on Line A. The Cipro stop is labeled as 'Cipro-Musei Vaticani', and the Ottaviano stop is labeled as 'Ottaviano-San Pietro-Musie Vaticani'.
The Colosseum is located off the 'Colosseo' stop, only 2 stops from the Termini station on Line B (the Blue Line) of the Rome Metro. There are actually great views of the Colosseum itself from right on board the Metro as you approach. The
You would also pay to board the Metro locally before boarding.