using an external monitor and a projector with a macbook air

There are lots of options but more information is required. For example, do you want to have different images on both devices, or the same image on both?

You need to provide details regarding the projector also. For example, most of them have HDMI ports.

An Apple TV works with AirPlay: How to AirPlay video and mirror your device's display - Apple Support

For external monitors in general, please refer to Use an external display with your MacBook Air - Apple Support.

What model of MacBook Air do you have? Apple menu > about this Mac, e.g. MacBook Air, Early-2015. Models prior to Late-2018 support only a single external display, so you cannot connect both a projector and a monitor.