I'd simply copy the stuff you need to a USB stick and forget about reusing the old hard drive.
However if you're determined to do so then, as has been said, using a caddy to turn the internal drive into an external (USB) one is the way to go. You can buy them for under a tenner but this one accepts both IDE and SATA hard drives:
http://www.pcnation.co.uk/product.aspx?id=5249&af=11
What nobody seems to have mentioned though is that if you simply put your old drive into the caddy and connect it up with a USB cable, most of the files on the disc will seem to have vanished. That's because you need to 'take ownership' of it before you can access all of the data on it. See here for the instructions you'll need:
http://www.techfleece.com/2011/05/15/how-to-take-ownership-of-files-or-folders-or-an-entire-hard-drive/