Two Factor Authentication on 10.10.5 Yosemite I cannot find where to get my two factor authentication verification code from in settings on the Mac, neither is there any input field for me to type in the code. What do I do?
You need to be running El Capitan or later.

See Apple KB - Availability of two-factor authentication for Apple ID

If your Mac can run Yosemite, it can run El Capitan.