If you're planing on running Vista on the hard drive from the old computer you should reinstall once the hdd is installed in the new machine.
Windows doesn't like major hardware changes and it was never designed to be moved between two different machines. (Actually it has features designed to prevent just that.)
If you're just want to use it as a slave I still recommend you format it as it will still contain all the Vista system files and boot information (the boot information can only be removed by formatting the drive).
My new motherboard will have SATA connections on the back... can I use the old hard drive that way to pull data off if I have to buy a new HD?
Yes, If the old drive is a SATA drive.