Go to Radio Shack or some electronics store and get a couple of cables that
are 1/8" stereo male on one end (to go into your sound card) and left and right
RCA male on the other end. If you are running your turntable through your home
stereo, plug one end (RCA) of the cable into "Tape Out" left and right and the
1/8" end into your "Aux In" on your soundcard. Then use Audacity to record a
stereo file for each song and export it as wav or mp3 (wav quality is better but
mp3 is 1/10th the size). If you're going to burn them onto CD's to save, wav
would be the one. If you're just saving them to your computer hard drive, mp3
takes up a lot less space.
You can use the other cable to go from your computer headphone out or the output
from your soundcard to your Aux in on your home stereo for listening to music on
your computer on your home stereo.