The best resource that I have ever found for a really good introduction to gear design is the technical section in the
Handbook of Gears by
Stock Drive Products. You can download the whole thing as PDF files from the SDP web site. As good as the stuff on the Martin site is, they really don't get into the background knowledge you really need to understand what is going on and why the formulae work. The Handbook Of Gears is a bit surprising because it is tucked away inside the SDP catalog. But it is a great reference for understanding both introductory gear theory and application considerations.
There is a whole section (see
Section 13) on gear tooth strength and design. Most of their calculations are based on the Lewis formula, or a modified (Barth) Lewis formula. As has been discussed several times on these forums, the Lewis formula calculations are based on industrial duty cycles, and the assumption that the gears will undergo millions of rotations. For almost every FIRST application this will result in a very conservative solution. While these calculations will provide a workable solution, it will be a bit of "overkill" for most FIRST robots (and therefore, lead to a potentially heavier gear system than is really needed, as the recommended gears are larger - and heavier - than need be). Joe Johnson's suggestion of using a beam stress calculation to determine tooth size will provide a very workable answer that will be closer to the optimal solutions for FIRST applications.
-dave