If I moved after last tax season ended, what zip code should I enter on the first screen (the "let's start by getting to know you" page)?

If you are entering your address in My Info use the address where you want to receive mail from the IRS now.

Your current zip code. 

You will need to enter your most current information.

This ensures the taxing agencies can contact you should they need to contact you or issue you a refund if there are any issues with bank information.