I learn something every day. If you have any suggestions in this regard, I would very much appreciate you sharing your expertise!
It's very simple. Most PCs, even with builtin soundcards, have a line-in. In a pinch you could use a mic-in, maybe. You simply have to connect the (preamped) turntable to the PC with one of the now-very-common RCA-to-3.5mm patch cords.
Audacity (free audio editor) is very easy to use making it easy to edit out big clicks and pops; it also has denoisers and turntable speed-adjustment filters, so you can even record 45s if you turntable only does 33s.
Thus if you have a listenable turntable and PC you are already good to go. You either have a turntable preamp, or you are using a stereo amp that has a phono input. Any stereo amp with a phono input almost surely has "tape out" connections in the back; use those. If not any old crufty stereo amp from goodwill will. The stereo amp section doesn't even have to work, just the internal phono preamp.
If you have a bare turntable with no stereo amp, you (probably) can't hook it up straight to your soundcard because the levers are so low. You could always try though. Some turntables have built in preamps, but not many.
This website has pictures, but ignore the bit about software.
http://www.microsoft.com/windows/windowsmedia/knowledgecenter/howto/FromVinyltoComputer_StepByStep.aspxI have recorded many albums using the builtin sound on my HTPC connected to the tape outs of my stereo receiver, and the quality is very very good.