How to transfer an entire partition from one drive to another?

My moderately tech illiterate parents bought a used PC a couple of years ago, which contained an HDD and an SSD, but somehow they didn't notice the HDD and set it up entirely on the HDD, without even formatting the SSD, I discovered the SSD a couple of days ago, confirmed it works and I think it'd make sense to move the entire C drive over (they're both about 500gb, so everything will fit fine) to make it a bit faster. I've got them to back up everything but I'd still rather avoid data loss. My current idea was install Linux from a usb and do it all from there, shrink the c partition a bit to make sure it fits, copy it to another partition on the other drive then delete the original partition and delete Linux. Would that work fine, and can you guys see any potential issues, or software that would be broken by copying it over then deleting the original.