How should I go about transferring recordings from cassette tapes to my computer?

Just helped a client with this very question recently.

All you need is a pc with a line in jack (most modern PCs have one, it's usually a light blue jack), a cable with a stereo headphone plug on one end and either a stereo jack or RCA plugs on the other depending on the output jack type on your record or cassette player. When I did this I got a cassette boombox and a cable with a stereo headphone plug on both ends, connected it to the boombox's headphone jack and my PC's line in jack and recorded using audacity.

You can get audacity here:

http://audacity.sourceforge.net/download/windows

Don't forget to also download the LAME MP3 encoder on the same page just below the audacity link.

For a video howto for recording from vinyl or cassette to MP3 or CD, go here:

http://www.youtube.com/watch?v=TEJdG2vTuWE

Note: you can record a whole album and then split it into individual tracks afterward, or record the tracks individually if you have time.

For a howto on splitting the tracks apart, go here.

http://www.youtube.com/watch?v=L7WBe6wP7Rw

There are many more related tutorials on youtube.