Congrats on the new device!
If you are a registered user (created an account), your lists are safe and sound in our server. Each Out of Milk account has login credentials (the email address and password you chose when you signed up) that can be used to access your account at any time from any device running the app.
All you need to do is enter the email address and password you originally used to sign up at the Log in page. Be sure to use the Log in option rather than the Sign Up option or you may accidentally create a new account.
If you've forgotten your account password, simply select the Forgot Password option at the Log In screen and enter your account email. You must only use your Out of Milk account email for this option, not any other emails you own.