Quickest way is to get an external enclosure for the SSD, plugging it to the still "original" Mac, formatting it, then "copying" the contents of the internal onto the external. "Copying" in this context means cloning, which can be done using any of the shareware products such as SuperDuper or Carbon Copy Cloner, or simply by starting up the Mac in Recovery Mode and using Recovery's Disk Utility to Restore the internal onto the external drive. Once the cloning is done, test it out by restarting and holding down the Option key: both drives should show up in the list, choose the external.
Once you're satisfied it all works, merely swap the drives. Keep the original drive untouched in the enclosure for a few days till you're sure all is ok, then you can use it for backups.