The PC with the desired files invites and shares to the receiving PC.
Get on the PC with the desired files and invite the receiving PC.
Accept the invitation on the receiving PC.
On the new PC, set up the new folder to accept the new share. For example if the older PC has the shared folder called "Sync", create the "Sync" folder on the new PC. Make sure it's empty on the receiving PC. The new receiving folder must be empty.
Don't delete the default share, it seems to mess things up when deleted. Just leave it there unused.
Get on the PC with the desired files and share them with the receiving PC.
Accept the invitation on the receiving PC.
There are a few ways to autostart Syncthing. Using the Desktop Environment is usually the easiest.