Calling that stuff “recording studio equipment” is a bit of a stretch but here is how you would wire it together.

First, your soundcard only has two inputs so anything you record, like drums, would first have to be submixed into your mixer and then sent to your soundcard. Also, your mixer has an eq, compressor and efx so why not use it. You’ll need that compressor for the vocals.

Your board doesn’t have any Aux sends or busses so you cant really use it listen back and record at the same time. Your soundcard, though, will allow you to monitor back and provide you with some basic headphone monitoring. That said, plug in your speakers to the audio out on your soundcard.

Note: do not get frustrated if the results are less than spectacular. The equipment you’ve acquired is great to learn on and you are gonna have hours of fun with it but it wont allow you to make a professional sounding product – sorry. It would be like trying to paint the Mona Lisa with a tennis shoe. Once you figure out how all the stuff you have now works, move on to some better equipment.

good luck