Mixing fonts is like mixing patterned shirts and ties—there aren’t immutable rules. Some people have a knack for it; some don’t.
Bear in mind these general principles.
The rule of diminishing returns applies. Most documents can tolerate a second font; many fewer can tolerate a third; almost none can tolerate four or more. (If you’re making a presentation, treat all the slides as one document.)
You can mix any two fonts that are visibly different. If you’ve heard you can only mix a serif font with a sans serif font, it’s not true. Much like mixing colors, lower contrast between fonts can be more effective than higher contrast. Look at any American newspaper—typically, the body text and the headlines are both in serif fonts, but different ones.
Font mixing is most successful when each font has a consistent role in the document. In a research paper, try one font for body text and one font for headings. Or try one font for things in the center of the document (body text and headings) and one font for things at the edges (line numbers, footer, and other miscellany). Or in bulleted and numbered lists, try one font for the bullet or number and one font for the text of the list item—a technique I use throughout this book.
It rarely works to have multiple fonts in a single paragraph. Better to restrict yourself to one font per paragraph, and change fonts only at paragraph breaks.
While I’m reluctant to endorse rote methods, this one usually works well: mix fonts by the same font designer. For instance, pairings of Atlas and Lyon (both designed by Kai Bernau), Alright Sans and Harriet (by Jackson Cavanaugh) or Concourse and Equity (by me).