The train station runs from the airport and is much cheaper than a cab. At the airport there are small stores, kind of like a 7/11 that sell the train tickets Ask someone to help you with this because it can seem confusing.
There are several stops so stay alert to where you want to go.
THere is one stop that lets you out about 150 yards from Vatican city. There are also public buses from the airport that take you most anywhere.
NOTE: Beware of pick pockets! On the bus this is at times a problem and usually it is from children or nice looking women. The locals, like in most of Europe, do not value personal space the way most of us do.
I say this because not everybody who gets close and friendly is a thief.
For a good book to use for sight-seeing, I like the Eyewitness books. They have walking tours and tips for budget travel.