Software update

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence.

Computer Qualifying for macOS 13 Ventura 

Computer Qualify for the Monterey upgrade

macOS Big Sur 11 hardware requirements

macOS Catalina 10.15 hardware requirements

macOS Mojave 10.14 hardware requirements

Make Time Machine Backup  is very useful 

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur, Monterey, Ventura supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may require an update or upGrades to 64 Bit Version

You need a 2012 or later Mac to upgrade beyond High Sierra 10.13. What year is your MacBook Air?

Just to note, the difference between Update & Upgrade, 10.13. can only be updated to10.1.2.6, but to go beyond you need to upgrade to 10.14, 10.15 etc.