Vista (released 2007) is recent enough that the drive is
probably a SATA drive. The connector will look like the one at the bottom of this pic:
If it's SATA, then you can just plug it into your new computer. Windows 10 should be smart enough to boot off the correct drive, but if it tries to boot off your old drive, you can hit esc or F10 or F11 or F12 to get a boot menu, and select the correct drive/OS to boot. After you get back into Windows 10, your old drive will appear with a new drive letter, and you can just copy the files you need onto your new system. Heck, you can leave the old drive plugged in if you want. It'll be slower so you won't want to use it for anything speed-critical like games. But it would be fine for storing movies or backing up your SSD.
If it's an IDE drive, then your job is going to be harder. You need to either find an old computer that still works and accepts IDE drives, plug the HDD into that computer, and copy the files over the network or onto a USB stick or external drive. Or you need to find an old IDE to USB external enclosure and put the drive into that. There are also IDE to SATA adapters, but I haven't had very good luck with them.