I'd like to start off by saying that the walk to the Sistine chapel is a long one. First you buy your ticket and climb the main steps, then you go through hallway after hallway until you reach the chapel.
The Sistine chapel is worth the walk though!
On the walk you can pass through the lovely Raphael rooms. They were commissioned by Julius II in the early 1500's and Raphael and his workshop covered the walls in frescoes. This is the School of Athens.
Not only did Raphael paint ancient Roman philosophers, but he included a portrait of himself and Michelangelo. Michelangelo is the man in purple in the foreground. Raphael and Michelangelo were working around the same time, so it is likely that Raphael would see Michelangelo sketching plans for the Sistine chapel.
The Michelangelo that Raphael painted is very muscular and is similar to the sibyls and prophets Michelangelo painted on the Sistine chapel ceiling.
Here is a photograph of Jeremiah by Michelangelo.
(http://www.italian-renaissance-art.com/Prophets.html)
The Vatican museums also houses a collection of modern religious works by artists like Diego Rivera, Francis Bacon, and Jacob Lawrence. This one is by Botero. Now, many people don't like his work, but I think this piece is just really fun. Everything seems to have been stretched, the priest, the trees, even the chubby cow.
The collection of Egyptian art is fantastic! I really like these urns.
The Belvedere courtyard was originally planned by Bramante, yet his project was built on top of and the existing lawn is much smaller than his original plan.
This is one of the most famous ancient Roman sculptures today. The entire collection of ancient Roman artifacts is really incredible.
The famous stairwell! The entire spiral staircase is covered in signs that say "Watch your step!" since each step gets bigger as you go down and the spiral closes.