I need to be able to log into my account without saying "Log in with Google Account". Is there anyway that I can log in with a username and password?
I guess when you created the account it didn't need a password as the login was through Google account. Check your settings page https://www.duolingo.com/settings/account. This has the username and the email linked to the account. I guess the email will be the one linked to the Google account. If not, you can update the email address here.
You can try logging in with the username and Google account password and see if it works. If not, you can go to https://www.duolingo.com/forgot_password and try resetting the password. This should send across a mail to the email address linked to the account with further details on how to set a new password. Once, you have the password, it should be simple enough to login with username and password.