I would get the cheaper Metrobus option of 10 tickets. You can use it from the airport to town and, once in town, you'll find that you will spend more time walking than going on the Metro. Most tourist sights are relatively central.
Transfers between bus lines and between a bus line and Metro are not included in single tickets nor in 10-trip tickets. They are only 'free' if you have any unlimited trip pass, like the tourist cards, or the monthly passes.
The 10trips ticket will allow you to go both on bus and subway (airport included) but anytime you cross the door counts as 1trip. The system does not include the notion of being valid for a certain amount of time to allow transfers. It is intentional and is a good way to keep it simple.
First of all, click on the 'English' link at the top of the page. Then, use the 'Recommended path between two points of Madrid Municipality' link on the left. Select your starting point, your destination, and then a 'Calculate path' will appear bottomright.