I have tried to make a new Trainers Club Account, but the site is down. Does anybody know how I can use my Google Account to make an account? Thanks.
On the opening screen (after the Niantic logo screen) there are two buttons. The top one allows you to sign in with a Google account. Tapping it will present you with a pop-up menu of all the different Google accounts associated with your phone.
It's not the same color as the Trainer Club button, so it's easy to miss.
If you do not see the button, try uninstalling and reinstalling the Pokemon Go app, and when asked your age, saying you are over 13.