A new MacBook Air doesn't have an onboard ethernet port. You could buy an adapter and an ethernet cable but as you probably won't use them afterwards it's a better idea to backup the old MacBook using TimeMachine (every Mac has this application).
Buy an external hard drive and backup your old Mac (a first backup will take several hours)
https://support.apple.com/en-us/HT201250
You can decide later if you want to encrypt the backup disk or not (it takes quite a while)
When the Time Machine backup is ready, you can use migration assistent to transfer files and settings.
https://support.apple.com/en-lamr/HT204350
You'll need a USB-C to USB adapter to connect the external drive to the new MacBook Air
After migration assistent did its work, use the external drive to backup the new MacBook Air on a regular base.
You could face some problems: there probably are 32-bit apps on the old MacBook (e.g. older versions of Word or a non-compatible printer app ) that will not work on the new one. Start a new thread if you have questions about 32-bit apps