Two things I can mention. If you do decide to wax i strongly recommended get it professionally done. Male or female you should be able to find at least a few salons local to you that do waxing. Most places in my area will do you entire face (including eyebrows, neck and sideburns) for about $30-$45. The reason I say this is because the skin on you face (especially your mustache) area is extremely sensitive and improper waxing can damage you skin.
To clarify about shaving. Shaving does not
actually make the hair thicker or darker. It only makes it appear as such because the hair growing out of your face/body is not a brand new hair, but rather a blunt strand being forced up by your follicle. This is why people tend to notice that after they've waxed the hair that grows back in tends to be thinner than when they'd shave. The idea that shaving makes hair thicker, longer or darker is a myth. (source