To open the Developer Console in Salesforce Lightning Experience, click the gear icon at the top right and select Developer Console; it opens in a new window. In Salesforce Classic, click your name, then Developer Console. If the option is missing, the user needs the API Enabled and View All Data permissions, and the org must be an edition that includes it.
Below are the steps with a current screenshot, the exact permissions from Salesforce Help, how to give a non-admin access with a permission set, and fixes when the Developer Console is not visible.