Cheapest: buy a mic, plug it right into the audio input on your
computer and download Audacity (free). Quality might be good enough, it
won't be great.

Next: get something like a Zoom H2. You can use it as a mic and mic
preamp and go into the computer with a usb connector (I think, I
haven't tried this).

If you want to play along with BIAB, plug a stereo into the headphone
output, set up your amp, and put the zoom h2 somewhere near them both
and record. Use Audacity to edit the file, if needed.

There are some boxes that you can plug the guitar right into and then
plug the device into the computer.

Of these, I'd suggest the Zoom h2 or similar device (there are a bunch
of them that do about the same thing). It's not because it's the best
for this particular application, but as you start playing with other
people it is very useful to have a portable recorder

I have the zoom h2. It's okay. There are better sounding choices, I
suspect, like maybe the edirol 09. But the zoom may be the best for the
money.