Startup disk

I keep getting a notification on my MacBook Air 2014 that my startup disk is full. Do I just delete a bunch of stuff?

The short answer is yes, but start here: How to free up storage space on your Mac - Apple Support.

I am assuming my laptop is out of date, should I buy a new one?

It's far from becoming an antique, but I won't try to talk you out of it. If I had to guess, your Mac never had a generous amount of storage to begin with. The smallest option back then is considered insufficient by today's standards, so if you want to purchase a new MacBook Air keep that fact in mind.