PC... OK, good. You need to get the sound into the PC.
Microphone, direct from instrument, midi... You need an interface between these devices and your PC. A standard, cheap, built-into-the-PC sound card won't do. What interface you get then depends on exactly what you need. Many choices here.
Recording, editing, mixing software. Lots of good stuff out there. Some free, some quite expensive. While your editing, you want to hear the sound... yes? Monitors and/or headphones needed.
You could do your mixing in a hardware mixer, but you can also do it in the PC software. No external mixer needed. Personal choice.
Basically... microphones, or othen instrument input --> interface --> software That's the reeeealy simple version. Start by researching that. Lot's of reading to do here.
