Bitcoin core does not produce a wallet at all
Bitcoin core automatically generates your first bitcoin address for you but you can create as many new ones as you want.


This is outdated. An address is no longer created for you by default.

You can create an address from within the Receive tab by clicking the "Create new receiving address" button.


  Here is what I see:


You are looking at the Sending Addresses window which is for addresses that you want to send to. It is not for your own receiving addresses; that would be the Receiving Addresses window. In recent versions of Bitcoin Core, the functionality to create a new address in the Receiving Addresses window has been removed.