In the hardware department you need:
1) The best record deck you can get your hands on. There is no way to recover sound that the pickup never picked up.
2) A top notch pre-amp with an input to match your pickup. The very best pickups are (were?) known as moving coil. Moving magnet (aka magnetic) pickups can also be good. In my book the only high grade ceramic pickup ever made was the Decca Deram. All of these pickups require a different kind of preamp input.
3) Cables to connect the Aux-out sockets of your preamp to your sound card Line-in. Please don't use your amplifier speaker sockets!
4) A good quality sound card. I know they aren't all created equal but cannot really recommend one. Maybe somebody else can help.
5) Lots of hard disk space. Audio files use up 10 Mbytes per minute.
6) A bog standard CD writer.
In the software department you need:
1) Sound recording software. This may come with your sound card but it might not be the best. Windows Sound Recorder is a piece of junk fit only for voice recordings through cheap microphones. There's nothing very special about the software you need EXCEPT for the record level meters. It's acceptable - sometimes even advisable - to allow musical peaks to go 'into the red' on analogue recordings but digital is different. Going over the limit on an analogue tape recording causes a gradual increase in harmonic distortion. Doing this on a digital recording will cause clipping which sounds terrible.
2) Look for a shareware program called Wave Repair. It's exactly what you need for cleaning up those pops and clicks you always get from old LPs. It also has a very good set of record level meters and this part is freeware.
3) CD writing software. This usually comes with your CD writer. If you have none you can (or could) get Record Now free from HP, though finding anything on their poxy website can be a nightmare. WARNING: on my CD writer (HP 9150i) Record Now deletes two seconds from the end of the last track. You can get round this by recording an extra two seconds of silence.
And finally --
You will need lots of TIME. How much time you spend will determine how clean the final results are. If you want CDs that sound exactly like the original LPs complete with pops and clicks you can do the job quite quickly. A really good clean-up job could easily take a day's work per LP. Don't forget that LPs can also be cleaned up BEFORE you play them!
Hint: Record each LP side as a single WAV file then split it up into tracks later. Also, don't worry about synchronizing your recording software with the start of each LP. You can trim off any leading and trailing silence afterwards.
This website has loads of valuable information:
http://www.cdrfaq.org/