I would suggest getting an external USB 2.0 enclosure for your drive. It is probably IDE, but check first, because you'll need to select the right enclosure for the drive interface.
This will allow you to connect it easily, as needed, and also continue to use it for backups.
The easy way would be a USB enclosure. If you don't want to spend any more money, you can take the old hard drive out, set the jumper to slave(if its IDE), then connect it to the new computer (via IDE or Sata). Vista should detect it as a storage drive and you can transfer all your data off it.