First thing, old computer will have an IDE hard drive in it, new computer will have SATA hard drives (different connectors) so you'll have to check that the new computer actually has a connector onthe motherboard to plug an IDE drive into, otherwise your onto a no-go from the start.
as long as it does have an IDE connector then yep, pretty much just plug it in and you will see another drive appear in "computer" and you can then just copy any files you want across to the new drive.
If the new computer does not have an IDE port on it to connect the old drive then a USB caddy for the old drive is going to be your easiest route to retrieve the data from it.
Once you have the files you need you can format the drive and use it as extra storage on the new machine, but is it worth it? your new computer will probably have at least a 160GB (+) drive in it, the old drive is probably tiny compared to that!