To add additional columns in the Discover tab click on "Add" next to the required field on the left menu in Kibana --> Discover.
Once the field is added to the Discover tab, you can move or remove the field by pointing to the field name in the table and then you can click the arrows to move the field or the X to remove the field.
Once you have your columns configured to your liking you can click on the save button above the search bar. This will prompt you to save a search, the search also includes the column layout. So, later on, you can open the saved search to see the layout you have created.