I have some experience with Logic Pro and Protools. The main obstacle with each is that both really need some type of audio interface to fully utilize their potential.

I would divide DAW and audio interface into software vs hardware. The DAW is the software (sometimes linked to specific hardware) that you use to track and record music. (e.g. Protools, Logic Pro, Ableton, Garage Band to name a few). The audio interface is the hardware that translates your analog instrument into a digital recording. Sometimes the instrument and the interface are one and the same. (e.g Ableton Push, a digital mixer or a digital keyboard controller). To record, typically you need a DAW, an audio interface and an instrument, microphone, or keyboard controller).

Your mixer likely will work as an interface if the USB lets you connect to a computer. DAWs come in all price ranges and configurations. If you are a student or teacher there are big discounts. For example Protools studio is $99 if you are a student. If you have a Mac, you probably already have Garage Band for free. Logic Pro also is Mac oriented but costs a couple of hundred dollars. Once you get a DAW, I am more than happy to give you a quick tutorial.