Good place to start.
Note: If one of the computers is running Windows XP, you might need to put the computers in the same workgroup.
To do this, on the XP PC go to:
Start >> Control Panel >> System. Select the Computer Name tab. Click the Change button. Change the Workgroup name to the same as your Vista PC (I think the Vista default is just WORKGROUP). Reboot.
You might also need to turn on file sharing on the computer running Windows XP.
Once that's done, do this on the new Vista PC:
Click Start, and then click Network. The Network window displays a list of computers in your Workgroup, which should now include your old PC.
Double-click the name of the computer containing the shared files/folders that you want to access.
Drag & drop all the files you want to your new PC, and VOILA.
If you have problems with any of this, state specifically what they are please, rather than say "I failed" because then nobody has any idea why! :)