WordPress submits search, by default to bloginfo( 'wpurl' ) with search parameter of s set to the search term.. You should be able to do this yourself. You will need to use GET on the form and set name="s" in your search field.
That said, I agree with the comment above by @Howdy_McGee that you should be using get_search_form(). You should just apply custom CSS to this to achieve your desired appearance. Your search form can’t do anything that this form cannot unless you’ve extended the search feature of WordPress.