When you're speaking with the other pageant contestants backstage, be gracious and kind. Your rivals can help or hurt you in a hundred different ways, from pointing out when your slip is showing to "accidentally" stepping on the hem of your dress. Be sincerely nice because few people are more adept at spotting fake niceness than beauty pageant pros.
While the swimsuit competition is usually unforgiving, the evening wear portion of the event allows contestants with the right attire to play up their strengths and play down their weak spots. Get a dress that fits you and flatters your body. Bespoke dresses cost more than off-the-rack numbers, but the investment is usually worth it, especially if you're entering more than one pageant.
During the interview portion, judges tend to grade on coherency rather than quality of thought. Rather than getting jammed up trying to formulate the smartest answer, respond to the question as you would if someone posed it to you in casual conversation. A simple answer delivered with naturalness and sincerity will generally receive far more points than a more sophisticated answer delivered haltingly.
When prepping for the swimsuit competition, some women go overboard on the dieting, dropping down to a stringy size 2. In most beauty pageants, judges still like to see curves on a woman. The ideal contestant's body looks full and healthy: sleek, soft and with just a hint of muscular definition. If you can see ribs through your suit, you need to start eating a bit more---anorexics don't win pageants.
Before entering the competition, do your research. Who is sponsoring the contest? What are the criteria by which judges are grading? Who has won the competition in the past and why? Doing your homework early will allow you to tweak your performance in advance to give you a competitive edge.