I'm not familiar with your equipment but it would seem your Edirol device is an audio interface that hooks up to your PC via USB and replaces your PC's internal soundcard (which, despite whatever claims are made by the PC manufacturer, generally isn't up to pro audio standards). I'm assuming that's what you're calling "soundcard" later in your post. It's more than a soundcard.. call it an interface.
Your mixer is of no great use this stage...
So you'd go
microphone (vocal or instrument) -> Edirol interface (the inputs are usually on the front) -> PC via USB -> DAW software
Your device probably has the ability to cope with both microphone level and instrument/line level inputs, so anything like a keyboard you could plug straight in. You may have to select an input impedance (mic/line) somewhere there to do that...
That enough for now? You're missing software at this stage. Reaper is a fully functional free trial and used a lot around here.
Read the other threads like yours.. there's about 5 a day... you'll find all sorts of links to useful info.