Hi Neil,
Here's what I do with no extra exspensive equipment, and I use free recording software. First you need to know how to connect your guitar to the computer. For electric, there are 2 ways. One: Run a chord directly from your guitar to the line-in port on the soundcard that is on your computer. You will need a 1/4" to 1/8" step down adapter to make the 1/4" guitar plug fit the 1/8" soundcard port. You can pick one up at Austrailia's equivilant to Wal-Mart or
Radio Shack. (Sorry, I don't travel much) Two: Run your electric through any effect pedals and then into your amp. Then run a cable from your amp's line-out to the soundcard's line-in.
For acoustic you will need a mic. and you can just plug the mic into the mic-input on the soundcard. (Note that there are several options for micing your computer that involve pre-amps and mixers and other expensive toys, but for beginners going strait to the soundcard is fine.)
As for the software side of it, I use Audacity,
http://audacity.sourceforge.net ,but that is only one of many free recording and editing programs out there. IMO it is by far the easiest to use and to understand. I just started using Kristal,
www.kreatives.org/kristal ,which is a little more advanced than Audaciy but also more complicated to use. I still use Audacity to record and I just use Kristal for advanced effects if I need too. Because I could type all day on how to use Audacity, and I'm sure half the folks reading this have dosed off by now, I'll just give a few pointers that I wish I had known when I started...
1) Your computer has volume controls for the Line-in and Mic-in, and these are usually muted if you've never used them. Find them in the Contol Panel and unmute them!
2) Audacity also has a main volume control and a volume control for whatever input you are recording from. Next to that control is a drop down box to select mic, Line-in, sterio mix, etc. Make sure you select the right option in that drop down box for what you are doing.
3) Finally, In order to save/export your recordings as mp3's then you will need a lame.dll plugin/decoder. You can get that file easily enough from the audacity site, but-and I can't say this enough, READ THE DIRECTIONS CLOSELY!! If you just click "download" and "ok" w/o paying attention to where it goes you will be lost.
Hope this helps, if you need anymore info about Audacity there is a topic on recording software that has a lot of answers.
James