Hi Mike, you have come to a good site to learn more about saws, I had cut firewood for over 40 years and didn't know anything about pitch, gauge or different chain types, just always got whatever was stock for the saws whenever replacing chains or bars.
the tooth count on a sprocket is somewhat analogous to gear ratio in an automotive rear end, in that a Higher tooth count will give a faster chain speed, but with less torque similar to a Lower Ratio Gearing increases a cars speed, but lowers torque. the pitch relates to the length of the links, your sprockets and sprocket nose bars will be pitch sensitive, but most sprockets are not Gauge specific, but the bar itself will be. Gauge refers to the width of the guide groove on the bar, and the corresponding drive teeth on the chain links. A smaller gauge chain would fit into a larger gauge bar groove, but it would be sloppy side to side and would result in the saw not cutting plumb.
I'm glad to hear you like the Timberline Sharpener, I have two of them, but I haven't had the time to really get used to using them, but the chains that I have sharpened with them so far, have cut extremely well, I was very happy with the edge it gave.
You will find a lot of Friendly people here(Just stay away from the Pol/Rel forums
) who are Happy to help each other out with problems, and sharing knowledge
Doug