i need an OS on this MACbook Air

OK, so assuming I've downloaded the version I want (or the one that seems most likely to take on my MacBook Air), and that I've been told the installation is successful, I figure I want to boot into the new OS. Don't see that in the instructions. When I restart, I am still in Mavericks.

Read the hardware requirements for each OS from the page you linked from and decide which version you want and then follow the instructions to download.

Welcome to the Apple Support Communities.

What year is your Mac?

Jack