I don't suggest shaving, its too much of a hassle and it doesn't grow back thicker usually, thats sorta a myth, but it does appear to be thicker because the razor chops off off the edge and it makes the hair appear jagged and sharp.
The best method that worked for me is sally hansen wax strips. Their only about 4 or 5 dollars for a box, and they have a bunch of them in there. Their just these strips that you rub together to heat up the wax, peel it off, and wax the moustache right off lol.
It definetely works for me, and I've tried a lot of things lol.
They sell them at most drugstores, or places like walmart.