If Old PC has USB ports (New PC certainly should have them!), then a USB memory stick is quite cheap and holds a large amount of Stuph. Like, about £25 for 512Mb, or cheaper. 1Gb sticks come in regularly at around £35 some places. Copy a stick-full over, erase what's on the stick, and copy another stick-full. Shouldn't take too long.
If it's got a DVD/CD-ROM ReWriteable drive, it should have software to go with it, but the newer versions of XP allow you to do it through Explorer. When you load a blank CD (and, I assume, DVDs, but I haven't got a DVD writer, just CD) you get a pop-up window asking what you want to do with it. There's an option which allows you to use it as a blank 'drive' - I forget exactly what it's called, but it should be fairly obvious if you read the options - and then you simply open Explorer, drag files to that 'drive', and then there's a menu option to 'burn files to disc' or something similar. Easy-peasy.
Ttransferring files
If it's got a DVD/CD-ROM ReWriteable drive, it should have software to go with it, but the newer versions of XP allow you to do it through Explorer. When you load a blank CD (and, I assume, DVDs, but I haven't got a DVD writer, just CD) you get a pop-up window asking what you want to do with it. There's an option which allows you to use it as a blank 'drive' - I forget exactly what it's called, but it should be fairly obvious if you read the options - and then you simply open Explorer, drag files to that 'drive', and then there's a menu option to 'burn files to disc' or something similar. Easy-peasy.