So if I were to acquire Windows 7 Home Premium which came with the laptop I want to get back running, and install that OS on the functioning hard drive, I might be able to just plug the hard drive into my newer laptop and have fun? Or is it very surely that it'll work - I'd prefer that.
1. Back up all data off the old laptop's hard drive (otherwise all data will be lost)
2. Put old laptop's hard drive in new laptop.
3. Using the recovery discs you made during laptop setup, reinstall Windows.
3a. In the event you didn't make any recovery discs, locate the Windows Product Key. All laptop manufacturers are required to put it somewhere on the laptop. Then download the correct ISO for Windows 7 Home Premium (must be the same architecture) as the one the key corresponds with http://www.mydigitallife.info/official-windows-7-sp1-iso-from-digital-river/ Burn that to a DVD and reinstall Windows 7 using the product key on the laptop.
4. Reinstall the drivers your new laptop needs from the laptop manufactuer's product page for your laptop
5. Reinstall all your applications
6. http://www.howtogeek.com/howto/4241/how-to-create-a-system-image-in-windows-7/ to streamline the process in the future (http://redobackup.org/)
7. Restore your data backup.