I just created an organization but it isn't giving me any options to create boards. I tried creating a board in the default organization and moving/copying it over, but it doesn't give my new organization as an option in the pull-down menu.
Creating a board can be done via the plus-button in the upper right corner (assuming you're using the web UI) left to your profile picture and name. When creating a board, you can specific an organization under which the board should be created. If you don't have an organization yet, you first need to create an organization via "Add organization" in the same menu.