Your answer is simple, you need something like the Pre-Sonus Fire Studio, FireWire computer audio interface device. This provides 8 class A microphone preamp and/or instrument/line level inputs. That device is approximately $500 US and includes a very nice multitrack audio & MIDI software package. A great bang for the buck. If you need more than 8 simultaneous inputs, you would need a second unit. Unfortunately that would exceed your budget. Some analog & digital mixers have the ability to output a multitrack signal via FireWire of 16-24 inputs but you're talking $1500-$3500 for one of those. So that ain't doable on your budget either. Of course you would also need a desktop or laptop computer with a IEEE 1394 FireWire port. Numerous adapters are available if your computer does not have a resident FireWire port. Only one of my desktop computers has a built-in FireWire port and the others required FireWire PCI cards. My primary laptop had a 4 pin FireWire connector but I also purchased an external FireWire card I could plug into the laptop which had 6 pin FireWire jack which I also wanted. So in your budget, that's about it.
You'll be tracking soon.
Mx. Remy Ann David