After I updated Pokemon Go, I tried to login again with my gmail account. Every time I type in my account name and then my password, but after I hit the sign in button it sends me right back to the account name screen. Is there anything I can do to fix this?
Yes! I had the exact same problem. If you are using an iPhone and safari as default browser this should help you:
Pokemon GO > Sign in with google > Compass icon (bottom right) > Allow > Open with Pokemon GO app as prompted and there it is! You'll be shown to the typical loading screen with Gyrados and that means it worked!
When you go into Pokemon go and it asks you to sign up again if you signed up with email then click Google in the options below when you the page stating they need to have access to your account. On the top right corner should be your email. If it isn't the right one then click the email and you can change it. Once you have chosen the right email you should have everything back.