1. You could just take the HD out of the old one and connect it to the new one.
2. Depending on how old it is, it surly must have a USB conection somewhere even if it is USB 1 so you should be able to use memmory stick, it just takes longer
3. If you have a router with 2 Lan conetions, get a Lan cable. Conect both PC to the Net and put up this program
http://www.teamviewer.com/index.aspx
and transfer what ever you want to the new PC. You will need to have an extra keyboard handy to boot up your old PC but it does not need to be conected to a screen or have a mouse after you have put the program up for the first time.
4. IF the old one has a DVD/CD writer just burn what you want on a CD disk and transfer what you want that way.
5. Boot the old PC up, see what is on it and depending on the size of material if it is under 2GB use this program
http://www.dropbox.com/
transfer to that, put up the new one, put Dropbox on it and there it is.