when i buy a series on apple tv, can I download it and watch on an airplane?

Yes, users may download purchased (or select subscription) shows onto Mac, iPhone, iPad, iPod Touch, or PC with iTunes.

(Re)download TV shows and movies that you purchased - Apple Support

HOW do you download a show onto a MacBook Air?

See: (Re)download TV shows and movies that you purchased - Apple Support

• For modern versions macOS (Catalina or later): use the download options in the TV app.
• For older versions macOS (Mojave or earlier): use the download options in the iTunes application.

Find out which macOS your Mac is using - Apple Support

Else please mention which show you intended to download.

HOW do you download a show onto a MacBook Air?