brainsys:
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Best and cheapest solution is to hook up your old turntable/amp (if you have acces to one) to your PC. Tape output to Line-In may do the trick - though i have found a surprising number of PCs will only do mono. The same great Audacity software is free to download.
Otherwise get a decent quality USB turntable with a moving magnet cartridge (about £80/100), impress your elderly friends/relatives and recoup most of your dosh by selling it on when you have transferred everything (or earn a few bob/get access to great collections by providing a service to them).
Agreed, the preamp with a decent turntable would be my first option for optimum audio fidelity, although a bit over the top if you only have a few discs to convert. However, the selling point with this rig is the ability to handle 78s, which the pricier usb turntables will not. I would have killed for something like this a few years ago while my dad was still alive as he had a small collection of obscure shellac discs which will never see the light of day on vinyl, let alone cd. Back in the day, I stuck a mike in the trumpet of a wind up gramophone and recorded to cassette tape and he seemed pleased enough with that. This would have been so much better.