I'm curious as to how old your OLD PC is. :)
Old isn't the problem. It's simply the HDD is the bottleneck. Coincidentally, quick booting up and shutting down are one of the first observations you'll see with an SSD drive, even on an old computer. I've installed SSD's all modern systems, however the oldest systems would be on an AMD 8450E and atom based systems, they boot in a matter of seconds after the upgrade, some one Windows 7, others on Windows 10, the OS doesn't make a huge difference.
Also, most SSD's ship with clone image software, so you're keeping your old system image AND the old HDD is now an on the shelf static backup.