How do I add more options to the post-new.php page?

The topic is vast, so I’m not going to discuss every detail. In WordPress, there is a concept called meta box. Publish, Page Attributes, Featured Image these all are meta boxes. And the good thing is you can create custom meta box. To know how to do that you have to be familiar with add_meta_box() function which is one of the many meta box API functions and also the add_meta_boxes action hook. Here’s an article which will help you to get started https://www.sitepoint.com/adding-custom-meta-boxes-to-wordpress/