I have been very happy with a product called audiotools
is the web address. It records and processes directly (if that's the right way of phrasing)and so doesn't require extra memory. Note you will need 10 mB per minute of music. So for a 74 min CD you'll need at least 740 mB spare on you hard drive - which isn't too bad these days I suppose.
I connect the tape output from my amplifier to the line-in on my computer. If you try and connect your tone-arm leads directly you will need some sort of pre-amp.
Another programme which seems good is Cooledit. This also comes with noise reduction/scratch removal plugins. There are numerous scratch "fixers" - I've used popfix. One has to be careful though - saxaphone and other things can be recognised as scratches. The whole process can be quite time consumming. Also if the burning to CD-R goes wrong you end up with a coaster. I've been forced to burn at 1x speed to avoid limitations on my computer - so set it up and go away for 3 hours.
It is worth while - I've started relistening to many of my old lps.