Use the browser on your phone and go and connect to 'Google' and log into your account like you were accessing mail from the browser, which works.
If your account is setup with a 2nd Authentication method, when you enter your password for you e-mail login, it should flag you and open a box that indicates that you have a 2nd authentication method setup and it should ask you if you want a call or text message. Select text message and it will send you a 6 digit authentication code that you will have to use the 'home' button so that you can get out to see it and then the 'home' button held for 3 or 4 seconds to open the list of applications so you can get back to the 'Google' page that should be open for the 'Authentication Code'.
Once you enter the code, the Play store should work just fine on your Note 3 and you should not even have to authenticate to it at all.
Have fun, and I hope this helps.