PITCHING - Don't snap the ball, don't stare at the umpire, and don't vocalize your thoughts.
Umpires behind the plate generally fall into three categories: completely objective, friend, or foe. The very best umps are completely objective throughout the entire game. Others start the game unbiased, but fall into the latter two categories by game's end.
As the pitcher, you always want the umpire on your side, or at the very least, not against you. That is more important to you than anyone else on the field. Consequently, follow this simple guideline. Do not show up the umpire. Do not snap the ball back from the catcher, do not stare in at the umpire, and do not question his calls out loud from the mound. Good umpires understand that a pitcher feels frustration if he thinks he made a good pitch and doesn't get the call. They also recognize that they miss calls from time to time. What they won't tolerate is a player who becomes animated or verbal in protest. It certainly won't work for you, and could work against you as the game progresses.
This does not mean that an umpire will go out of his way to stick it to you. That is not their mission when they officiate a game. It's the 50/50 calls that will fail to fall in your favor. While umpires strive to be objective at each game, they are human beings underneath that mask. A pitcher who carries himself with maturity and doesn't complain throughout a game will likely get those close calls. The umpire will respect the fact that he's been shown respect. A pitcher who acts out is subject to an ump who takes his antics personal.
There may be a time when you feel the need to ask the umpire a question about his strike zone. The first option is to have your catcher ask him between innings. The catcher is hopefully building a personal rapport with the umpire during the game, so he is the person best suited to be your liaison. A coach may also inquire between innings, provided the coach understands the right way to approach it.
The final option is to ask the umpire an open-minded question to the umpire between innings. Television cameras unfortunately go straight to commercial when an inning is over. If the cameras followed Greg Maddux or Tom Glavine walking from the mound to the dugout after the third out, the picture would (occasionally) show the pitcher asking the umpire a question about the strike zone. Part of the reason veteran pitchers do it in this fashion is because they know the cameras are not on them, meaning, they're not questioning the umpire in front of a national audience watching at home. Umpires appreciate this. If you do speak to an umpire, remember to check your tone at the door.
The best mindset is to concentrate on learning the umpire's strike zone. Each guy behind the plate will be slightly different. Adapt to his interpretation of the strike zone (up, down, slightly off the plate or slightly in) and reap the benefits. Fight his judgment and your frustration may adversely affect your performance.
One final note. Umpires WANT to call strikes. There are not too many umpires that you'll talk to before a game that say, "I hope this pitcher is wild." They like to call strikes. Appease him by learning his strike zone.