You can buy a turntable with a USB connector (
google for it).
You can connect your stereo to your sound card input, and use a free audio recorder / editor software package, such as
Audacity, as suggested above.
Or you can use a different software package such as
Goldwave. (Which I recommend, if you don't mind the price.)
Once you've recorded the sound (plus a few seconds before and after, or even the whole album), you can easily "select" the individual songs and save them separately in ".wav", ".mp3", or other popular formats. If you want to get fancy, you can even apply filters to try to reduce imperfections (hiss, etc).