So in my years of doing graphic design I have come across many fliers, business cards, brochures, etc that are, well.. inundated with many font styles. Immediately I want to put it down and walk away and never see it again upon noticing this. Sometimes people think that loading as many different fonts as possible onto their flyer is designing it and will be more noticeable. Well, it's noticeable alright but for the wrong reasons...font overload!
There are no "rules" per say on the amount of different fonts you can use on a brochure or flyer. But the general consensus is no more the 2 different fonts. One way to accomplish this is to pick a non-traditional font such as Curls, Disney font, or something that has "some flair" for a header. This is what will get the initial attraction of the viewer. You want it to stand out from the rest of the content. However, this can also be done by using the same font as the rest of the content but in a different style such as demi-bold, bold, or black. Again, this will help attract the viewer.
Generally, the content should be a font that is easy to read. Another rule of thumb is the serif fonts help the eye flow from one letter to another when there is a lot of text such as in a book. But a serif font is not always necessary. Typically a sans serif font can be used also. One type of font to definitely avoid when considering for content is a font such as Gothic style fonts (see example below). Just because it has lower case letters available does not mean it should be used as content text.
Below you can see examples of how the gothic font would make reading a paragraph more difficult than a sans serif or serif font.
So in short... keep your fliers and brochures down to a minimum of two different fonts. Different font styles such as Helvetica Light, Helvetica Semi-Bold and Helvetica Black are ok because they are part of the same font family. Keep your "fancy" fonts restricted to headers. Keep your main content font simple and easy to read and you will be sure to keep your prospective customer or client interested.