The software doesn't matter to the quality of the recording. Any program written in the last decade can be set to do 16-bit 44.1 KHz recording (which is what you want to start with.) Audacity is a fine program to use, but, again, it's not the software that matters.
Bons recommends
this mic. You'll need to get a 1/4" to 1/8" adapter but you can find that in a Radio Shack or equivalent, or buy it from the site as well.
You'll also want a windscreen, either from that site or making one with pantyhose and a coat hanger as described in Wynne's voicing tutorial (which you've read, right? No?
It's here.)
Unfortunately, the audio-in circuitry on a lot of PCs and laptops just isn't made very well, so if you're getting a lot of electrical noise and hum you may need to purchase a new, stand-alone audio card. Fortunately, they're dirt cheap these days. Shutting off all lights and electrical equipment near the computer helps a lot, as does (in the case of laptops) recording in battery mode only.
Aside from that, I recommend skipping the blanket and finding a way to set yourself up in the corner. A boom mic stand works well for this purpose (that site should have one for under $30), and you'll need to find a small table or stand to put the laptop on if you don't have someone to assist you with the recording.