I wouldn't recomment you to use Google Authenticator, since you can't easily backup your accounts with it and it's not tied to yur gmail account. So if you lose your phone, your 2FA accounts are gone if you haven't saved the secret code. I suggest you to use a open source app like andOTP (the one I'm using) that allows you to create a backup of your 2FA accounts.

Basically, when you want to login to your account, you will need to get your 2FA code from your phone and use it as an "additional password" (which is always different from time to time and it's only valid for one login session). This makes it very hard to lose your account to hackers, since they would need to breach your PC and phone at the same time.