First to try you could use your existing turntable and amplifier and soundcard to see
if you're happy with the quality.
Just connect a tulp/cinch to jack 3,5mm cable from a tape out on the amplifier to the line in on your soundcard.
If you're not happy with the quality you can always go for a better sound card or a usb turntable.
A great free program to record and edit is Audacity: Audacity: Free Audio Editor and Recorder
As Borg 386 say's you can record in MP3 format, the higher the bitrate the better the quality, or to get even better quality you could record directly to Wav files.