When I download the app Pokemon Go on my other phone, it just says "sign up". How can I log in with the same account on this phone?
When you first open the app, after entering your date of birth, the login screen shows with "Sign up with" and the choice for Google or Pokémon Trainer Club.
If your account is a Google account, simply press the Google button and select from the list of accounts. Your account will have to be already added to the new device. When you select the account it will automatically log you into the game.
If your account is a Pokémon Trainer Club account, you will need to enter your username and password and click "Sign In".
Note that these steps are for an Android device. An iOS device may have slightly different steps to log in.