Forgot my Google password but I'm still logged in on my phone

If it's an iPhone, don't sign out of the Gmail or Google app while you do the recovery. Google often sends the 'is it you trying to sign in?' prompt to that app, and tapping yes there is the easiest way through.

Android? Settings > Google > Manage your Google Account > Security > Password. Because the phone is a trusted device it usually lets you verify with the phone's screen lock instead of the old password. On an iPhone it's less reliable, you'll probably end up in the normal recovery flow.