Gday Dani,
If you want to record individual sounds into your computer, you need to bring them in seperately.
By plugging them into a mixer you are..yes...mixing them. So you wont be able to balance them seperately after the fact.
Protools is probably one of the easiest solutions IMHO -
a) lots of training options if you need them
b) both the hardware and software come from one company, all in the box
c) much more musician friendly these days
But - you are flirting with a totally different job. Audio engineering. No longer being a musician.
If you really need to go this way do it, but please understand it will be a fairly hard climb up the learning curve unless youve got mates around you with experience, to assist.
So what you're looking for - an "audio interface" or external sound card, that has enough inputs to cover all the bits you want to record AT THE SAME TIME - ie if you want to record drums (kick, snare, overheads thats 4 mics) plus bass pluss two vocal mics you'll need an audio interface with at least 7 inputs, like a 003Rackplus.
http://digidesign.com/index.cfm?langid=61&navid=464You wont be able to split the individual signals from your ashton mixer, whoever sold it to you sold you the wrong thing as it doesnt have individual line outs from the mixer.
Try talking to the guys at factory sound in south melbourne - they generally know what they are doing in the hitech dept.