Sometimes the screen will say “Couldn’t sign in” and you’ll feel personally attacked. Don’t throw your phone against the wall—I tried that, and it bounced off my sofa, which was terrifyingly anti-climactic. Instead, check your internet connection. Wi-Fi is your friend. Airplane mode is not.
If that doesn’t work, go to Settings > Accounts > Add Account. Yes, it’s buried deeper than a Secret of the Ooze. You’ll feel like a tech wizard when you find it. If all else fails, reboot your phone. Rebooting fixes 90% of life’s problems, including broken hearts and lost passwords.