Do we buy tickets to Vatican and Sistine Chapel asap and then go back
and get the guided tour as an add-on?
Nope. The tour includes your entry to the museums so you if you want the tour, book the tour and not the general-entry tickets. Just to note; the tour doesn't include St. Peter's; just the museums. There are tours through indy companies that do include the basilica, and it's not impossible that the Vatican might add that option down the road but not at this point.
If you haven't see the description of the tour, here it is.
"Full price ticket: € 40,00
Reduced ticket: € 28,00
Duration of the tour: about 2 hours
The tour follows an itinerary which includes: the Pio Clementino Museum, the Gallery of the Candelabras, the Gallery of the Geographical Maps, and the Gallery of the Tapestries (Renaissance art), the Raphael Rooms and the Sistine Chapel
Includes:
Entrance ticket
Guided Tour
Meeting point: On the day of the visit go to the entrance of the Vatican Museums - situated in Viale Vaticano – and show the booking voucher (electronic format allowed) to the Security Staff at the entrance (15 minutes before the scheduled time). After passing through security, proceed to the Guided Tours Desk to collect tickets and to meet the guide."
These tickets will sell out quickly once available for high-season months (fall is still high season) so crossing fingers that you land 'em!