What's the best way to mic or capture audio during a stage performance? Good question.
Let's start with some basics. Your choice of microphone will affect the quality of the audio. For most situations, a cardioid condenser mic will be the best choice. The pickup "reach" is much better than a dynamic mic and has less of a tendency to pick up unwanted audio from the audience. You also want to place those as close to the source as possible to minimize echo pickup. Unless you are recording a classical music or choral concert, then some echo is desirable.
Another frequently asked question is, how do you get those mics close to the action? For plays, I find that 3-4 cardioid condenser mics on stands right at the edge of the stage usually works well. You might want to keep them off-stage to prevent footstep pickup and accidental kicking.
Large bands and orchestras benefit from a wider pickup. Too close and you start singling out instruments. Omni-directional mics work best here as well as overhead mic'ing or a couple of shotgun-type mics in the back of the room.
Depending on their ages and genders and how they are staged, choral groups can benefit from both close and distant pickup. I tend to use a combination of condenser, dynamic, and PZM/boundary mics for this scenario. Placement depends on room acoustics. Many choral groups are placed on risers and thus at different levels. I usually place a couple of mics up high (at least at the level of the top row) and a couple lower (3-4 feet in height), and this is only if I intend to mic close (within 10 feet). Again, you have to make the call as to how much you want to use the room acoustics and what kind of audience noise will also be recorded.
Loud performances like brass or rock bands need either dynamic or high sound pressure condenser mics. Just watch out for proximity issues; placing them too close will change the audio "color" and make the sound too sterile, as well as require too many mics. Drums always pose a problem, as they tend to overpower the rest of the band. Ever wonder why bands on "live" TV shows have a Plexiglas wall around them? It provides additional control for mixing the drum level, but that is pretty extreme for event use. But the theory is good and usable. Placing a baffle or sound-absorbing material in the direct line between the microphone and drums will help limit the drum level.
Variety shows are difficult in that there is constant movement, as well as varying numbers among the performing acts, and everything from singing soloists to boisterous groups. It's best to attend the entire rehearsal and work with the people producing the event. If you're recording an event in which the individual talent uses body or handheld mics, that's great. But you should always be prepared for the worst and have backup—especially if young children are involved.
As for competitions, whether bodybuilding or cheerleading, always attempt to get a feed from the sound system (announcements are critical) and run a secondary mic for ambient sound and backup, just in case that direct feed doesn't work out like you planned.
Here is a little trick that I came up with when recording a soloist for a college recital. It was in a small venue (fewer than 100 people) and a raised performance area (not a stage but about 10 inches above the floor). I placed a cardioid condenser at the base of the riser pointing up at the singer and used a shotgun at the rear (about 30 feet back, where I was shooting from). In post, I placed the close mic at the center pan position (equal to both stereo channels) and the rear mic to the left of center (play with the position to get the desired effect). The result was pseudo-stereo that sounds good, even with headphones. This effect works with simple recordings (like the soloist) because of the time delay created with the 30 feet separating the two mics (sound travels only so fast).
Always, always monitor your audio, all the time. Find a comfortable set of "full cup" headphones (they aid in keeping out ambient sound) and keep them on. This is the only way you will detect problems immediately and be able to correct them before they ruin your shoot.
Play Copyright Clarification
Another area of concern is the question of copyright issues for plays. Forget the notion that you will find business taping community, semi-professional, or professional theater. That will not happen because of the way script and music score rental agreements are written. They all restrict both video and audio recording. School performances, especially for younger children, have (sometimes) a more liberal contract. Always check with the group producing the play to see the wording in their particular contract. For this reason, many elementary schools are looking at original plays and variety shows as an alternative. Almost all schools I deal with want a professional to record the event and are becoming increasingly upset at the restrictive contracts.
Other type of events, such as band, choral, and orchestra concerts, do not have the same recording restrictions, unless the recordings are to be re-sold specifically for the musical content, for profit. Then it would fall under the terms of a distribution/royalty license. That usually is done in a studio setting and outside the realm of event videography.