Hi Sara,
Normally, you should consider the branding of the product/service when picking the appropriate font for its website.
There is no “best” style that fits all because it really depends on how the site’s brand wants to be perceived.
You can learn more about fonts by checking this site: https://digitalsynopsis.com/design/font-psychology-emotions/#:~:text=Serif%20fonts%20portray%20tradition%2C%20sophistication,%2C%20classic%2C%20stylish%20and%20formal.
As for H1, H2, etc, We normally leave H1 for the page title. You can use H2-H6 for your article headings and subheadings.
Hope this points you to a better decision in picking a font.
Let us know if this helps.