An extremely simple, free for personal use recording software:
http://www.kreatives.org/kristal/
You essentially open the program, press "record" and then you're ready to go. It can do things other than just that, but they're not pushed in front of your face, so it's very simple to use for simple things.
Even a cheap modern electret mic sounds pretty OK, and if you just want to listen to how accurately you can play things, it'll be fine. I use a Rode NT1-A through a Line 6 guitar / mic interface (the NT1-A requires phantom power, so you can't directly plug it in with an XLR->3.5mm adapter), and even though the mic is hardly an expensive one (I think it was around 150e with a shock guard, pop filter, mic stand, cables, guide DVD, everything included), it sounds great. I've used it for vocals, acoustic guitar, accordion, sax, clarinet, ocarina and xaphoon.
IMO it's considerably better than my Audix OM2, or the Shure SM58, which is probably the most popular live mic ever manufactured, because it can double as a hammer and still work afterwards.