How do I change the style of just a part of the tagline?

A quick search online for html in your tagline came to this site: https://www.kirstencassidy.com/allow-html-code-in-wordpress-tagline/

It seems like it is your answer.

It explains that you would update your theme file to use the echo html_entity_decode function:

<h2 class="tagline"><?php echo html_entity_decode(get_bloginfo('description'));?></h2>

Then, you can just use html in your admin > settings page like so:

This is <span class="some-class">my tagline</span>