Use a tripod. Nothing screams amateur video like shaky, hard-to-watch handheld camera work. Pageants can go long, and even the steadiest hands will wobble after a short while. A well-placed and level tripod will allow you to operate the camera smoothly and focus on shooting coverage and getting good focus, while giving you consistent quality in your images.
Ideally, enlist a second camera to accompany you. Beauty pageants have a lot going on and have multiple participants, and an additional cameraman can help you capture all the action. If you instruct your second camera to capture a reasonably wide shot of the stage, what's called a Master Shot, then this will enable you, with the primary camera, to go in for close coverage. The best type of coverage is to move in on close-ups of the participants while they're performing, on judges while they're deliberating, and to get reaction shots of the crowd, the other participants and the hosts. As long as one of the cameras is always shooting the Master, there will always be something usable to cut to during the post-production editing stage.
Prepare a shot list ahead of time and reference it during the pageant. Make a list of what you'll need to capture on camera in order to effectively showcase the event, and keep this list on your person during shooting. It might seem obvious, but when the event is over and you realize you neglected to get very much coverage of the winner, for instance, you'll regret it. If there is a part of the pageant that you know ahead of time will be especially visually impressive, make sure you capture that.
Be ready for anything! While there is a loosely structured itinerary that all pageants will follow, it is a live event, open to last-minute changes, unexpected outcomes and acts of spontaneity from those involved. Be ready to roll with changes and go on a few unforeseen tangents. To some extent, you have to follow your instincts and trust that what you are doing is competent. Don't worry too much if a few things get past you; this is bound to happen with all that is going on.
Try to record clean audio for as much of the pageant as possible. Most cameras have decent mics that can capture the sound at the location, and most pageants will have some sort of PA system set up, but it might be worth it to rent an additional microphone to improve your filming. A condenser microphone, pointed toward the stage, or a camera-mounted shotgun microphone should be sufficient in most settings and will give you pleasing results, particularly if there is emphasis on music or singing during the pageant. Most rental houses have a huge selection of rental mics to choose from that can add production value to your video.
Be calm and keep your focus. Try not to panic or rush yourself, because when you're rushing, you're more likely to miss something, to film something poorly in a haze, to not have good focus or neglect an audio concern, or to just make a silly mistake. If you do make a mistake or miss something on the stage, get over it right away and keep filming. Lamenting one missed opportunity and then missing another because of it is counterproductive.