Visit to the Vatican

The Sistine Chapel is part of the Vatican Museums and furthest from the entrance. There is an admission fee. For an additional fee, you can prebook a timed entry to avoid waiting in line. Anecdotal evidence suggests that lines are short or non-existent after about 1 pm.
http://biglietteriamusei.vatican.va/musei/tickets/index.html

There is no fee for entry to St. Peter's Basilica, but there is a security line, which generally moves quite fast.

There is no fee for entering St Peter's Basilica, though there is usually a line to go through airport-style security,. Make sure you are decently dressed - no shorts, no bare shoulders or short skirts.
Siatine Chapel is part of the Vatican Museums, and you have to pay 14 euro to get in. If you are lucky, there is no line to get ion, but normally there is a long line - a few hours' wait is common. You can now book a timed entry ticket online at http://biglietteriamusei.vatican.va/musei/tickets/index.html (service charge of 4 euro will be added). You can book up to 60 days ahead.

Note that the two entrances are nowhere near each other. After you visit St Peter's - and I would be sure to visit the Treasury, climb up to the Dome and visit the crypt as well - you have to exit and walk all the way around the outside of the Vatican to get to the entrance of the Museum.

Or start with the museums, and after seeing the Sistine Chapel at the far end, use an exit there which takes you directly to St Peter's.