Be warned. Not only is the Sistine Chapel part of the Vatican Museums (so that's what you need to buy a ticket for), it's at the end of a long, Habitrail-like slog. And you're not the only one who wishes it were a "stand-alone" admission. This means mobs of people in the Vatican Museums are all just trying to get to the Sistine Chapel, and the tension and crowds are unbelievable.
Seriously consider paying extra for early or late admission. It costs more, but you see it with far fewer people, so it should be a much more pleasant experience. The Vatican itself sells some of these, as do private tour companies like Dark Rome and Walks of Italy. If you don't buy one of these, you definitely want to buy advance tickets to avoid at least that one line.
St. Peter's Square and St. Peter's Basilica are free to enter (can be a long security line for the Basilica), but you will need to buy advance tickets for tours of the necropolis. There are also limited tours of the Vatican gardens (look at their website if you're interested). Other than these and the Vatican Museum, other parts of Vatican City are off-limits to tourists. So last time, you didn't see "the Vatican," you saw St. Peter's Square and Basilica.