Add new post using a page inside the website

You have never read the WP documentation?

Create any template and form and when submitting the form it could be

Example:

$args = array('post_type'=>'post', 'post_title'=>'your title', 'post_status'=>'publish', 'post_content'=>'post content');

wp_insert_post ($args);

This will allow any post page or custom post type to insert the post