If you're running into any errors when you setup your account on your phone, please follow the instructions below:
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Open Gmail from a browser and only sign into the account you're trying to add. Be sure to be signed out of all your other accounts
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Go to this link: https://accounts.google.com/b/0/displayunlockcaptcha and click Continue.
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Open your phone's settings
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Scroll down to account and click Add account
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Choose Google>>Existing>>Sign in
If you have 2-step verification enabled on your account, you'll need to generate an application specific password for your account. To do so, go here, and click Settings, next to App passwords. Once there, generate a password to use for your phone.
If that isn't working, you may also need to enable Access for less secure apps. That option is found on your Security page of your Google account, under "Access for less secure apps."