The Vatican Museum is just around the corner from St. Peter's but it's a big corner. Often the waiting lines wind back to St. Peter's. Get there early!
Our primary destination was the Sistine Chapel. As you walk through the museum, there are signs pointing out the way to the Sistine Chapel. I believe that they run you all the way through the museum. There has to be a more direct way to get there. But it's worth the walk!
The wikipedia article is here and their web site is here.
We stayed in Welcome House at Via Nazionale, 230, just down the street from train station. There's even a McDonald's nearby! But we ate more gelato than Big Macs.
Their web site is here and their write-up on TripAdvisor is here. Many of the comments are in other languages. I think that gives you a feel that this is not a hotel that is routinely frequented by Americans. Remember that.