How to put a search box in my header?

You just need to add this to your header.php where you want to appear:

<?php get_search_form(); ?>

Unless you have a searchform.php file, wordpress is going to render this:

<form role="search" method="get" id="searchform" action="https://example.com/">
    <div><label class="screen-reader-text" for="s">Search for:</label>
        <input type="text" value="" name="s" id="s" />
        <input type="submit" id="searchsubmit" value="Search" />
    </div>
</form>

Btw, you should look the Codex.