If both systems and Firefox version are the same, you can use sync. Another way is to copy the old user profile to a thumb drive, then copy from the thumb to a new user profile. You may need to allow files to be overwritten.

I have Sync working so obviously that doesn't transfer the necessary settings. Where do I go to send my FF settings to a thumb drive?

Type about:support<enter> in the address bar.

Under the page logo on the left side, you will see Application Basics. Under this find Profile Folder. To its right press the button Show Folder. This will open your file browser to the current Firefox profile. Now Close Firefox.

This is your current profile. Copy the folder As Is to a folder on a thumb drive. Then Copy the folder to the new system.