You should be able to just sign in on the new phone with your old account. Since you appear to be having problems doing this, you really need to provide more explanation of what is happening that makes you think you cannot. We need more details in order to be able to help you. I'll start with some guesses.
This article tells you the various ways you can try to discover your forgotten Apple ID --> If you forgot your Apple ID - Apple Support Typically an Apple ID is an email address you have used. If you are currently using a mobile device that is logged into your account, go to Settings and tap your name at the top. Look in Keychain, and in a browser's security settings if you use it to automatically sign into Apple when using a browser.
This article outlines what to do If you forgot your Apple ID password --> If you forgot your Apple ID password - Apple Support
or go directly to this link --> http://iforgot.apple.com/
If you find you can’t sign in, reset your password, or receive verification codes, you can request account recovery (read this support article: How to use account recovery when you can’t reset your Apple ID password - Apple Support ) to regain access to your account. Account recovery might take a few days or longer, depending on the specific account information you can provide to verify your identity.
If you still need help this page provides country-specific Apple Support contact information ➞ Contact Apple for support and service - Apple Support