1) Shave completely,
2) Buy a false moustache in the style you want,
I've experimented with facial hair in the past, some might say I still am doing. I think the main thing is definition. Like already mentioned, split-ends and uneven growth make everything a bit fuzzy, so don't be scared to trim the surface. Also definition to the side. If you're not intentionally running your moustache around the sides of your mouth and into the cheek-pad parts of a beard, and also also melding into sideburns, decide where your limits are and be as aggressive as your particular shaving-methods can be in keeping the edges trimmed and the rest of the face stubble-free.
Also, from experience, you can sometimes prune a bit more than you wanted. A bit of wetting (after the whole shaving issue, whether that was wet or dry and 'combovering' can rectify that, if it's not too bad. I wouldn't advise (also from experience) hair-gel, although going the whole hog and waxing the moustache, a la
Poirot, would be a style statement.
I've never (intentionally) gone for moustaches/beards that grow beneath their 'bedding'. Like full on handlebar ones, spikey or curly pertuberances. Good luck with that, if you're going that way, eventually. Obviously you need to modify things from my plan to build up enough length to draw to the side and accomplish such things.
 It's alive! It's Alive!!! Mwahahaha! Argh! No, my creation, no! I am your master! You will
obey me! Arg... <gurgle> <thud>
 Personally, I dry shave with a (supposed to be used wet) safety-razor (edit: erm, disposable thing... YKWIM
), but I have an electric hair-trimmer for length control, of whatever bits I don't shave completely (and the head-hair likely to loll down in front of the ears, if I'm not currently sporting sideburns). Also useful when I let the stubble get a bit much for the razor.