The short answer is yes, there are multi-track recording programs. There's a freeware one called Kristal that is pretty good. I prefer to use Cakewalk's Home Studio, but that costs $100. However, most computers freak out when there's two soundcards in the system. It often doesn't work. Besides, if you plan to record with your sig rig, it will get bogged down very quickly. I do sound recording as a side hobby and also at school. Last night we did an acoustic night where we record 8 tracks simultaneously. That took at least 4 GB of hard drive space for 2.5 hours, because it records each input as its own seperate wave file. That system is a Pentium HT 2.8GHz, and it has nowhere near enough RAM for us, but I think it has like 128 MB.
You'll want a USB, Firewire, or PCI audio recording card. Good examples are pretty much anything from M-Audio (don't bother with the audiophile series), Edirol, PreSonus Firepod and Firestudio, and if you don't mind tweaking a lot, Alesis's Multimix 16 Firewire edition is a pretty nice setup. It's a mixer, but then it also allows you to record all 16 inputs at once over firewire, and all for only $600. But you will definitely want a better rig.
Also, most of those recording boxes I mentioned only have 2 or 4 mic inputs and only 1 instrument input. The rest are just line-level. You'll have to plug your instruments in via the direct out on your amps or just mic the cab if you can.