Save your Google Auth codes when you set up 2FA, it is only a question of time before your phone breaks. If you did not do it, disable 2FA now and reenable it right after, this gives you a second chance to save the key.
Couldn't you please describe how can I do that, assuming I have full access to my account? What codes are you talking about?
It's impossible to do it now. That's the nature of Google Auth, you have a chance to save the code only immediately when it is created, to prevent anyone from getting your codes while incidentally get hold of your phone for a short period of time, for instance while you are not watching.
Procedure to get the code is to disable 2FA on a site, get new authentication token for that site and then log the code. There's also a possibility to get maximum 10 recovery tokens on Google site, I'm not quite sure how it works since I was never in situation to use those recovery tokens.
What do you mean by logging the code? On bitfinex 2fa settings page I can see a QR code and an "Accont token(key)" which is a string. Do you mean that I can for example print this page, and when I loose my phone next time I will download a google authenticator app on my new phone, feed it with the QR code that I printed and It will work as the old one?