Here are my tips for creating better dance videos, particularly of performances.
1. Keep your camera as steady as possible. Use a tripod, monopod, table, or brace yourself in such a way that your hands don't move. That should be your goal. There's nothing worse than making your audience seasick. Let the dancers move around the frame a bit, and don't try to follow them left and right with every step as this will result in jerky camera action.
2. Keep the dancers in the frame. Give them some room in the frame to take a few steps, then if it appears they are going to walk out of your frame, follow them smoothly. When the dancers are far away from you in a large room, zoom in GENTLY to follow them, and zoom out again when they move close to you. The object is to keep them as large as possible in the frame without chopping off their upper or lower extremities. When they dance so close that you can't zoom out any further, you'll have to decide which half of their bodies you need to focus on: the embrace or the feet. If it's a fast milonga, you'll probably want to tilt down to their feet; if it's a slow tango with a particularly passionate embrace, it might be better to shoot that, at least until the dancers move away and you can get their entire bodies in frame again.
3. If you must zoom, do it gently. Practice with your camera in advance, to make sure that you are able to zoom in and out slowly, with control.
4. This might sound obvious, but remain silent while you are shooting. Other audience members might be applauding, voicing their approval or commenting on the dancers, but you should not. Your mouth is very close to the camera's microphone and you can't edit out that kind of thing later without losing the sound of the music.
5. Try not to pick up your camera right after you've been dancing. I have learned the hard way that a cold camera in my sweaty, post-dancing hands quickly leads to the lens fogging up! It will look like the dance floor is filling with smoke. Not a great effect. So if you have any advance notice that the performances are about to begin, excuse yourself from the dance floor and get set up.
6. See if you can get the lights turned up. Most milongas will turn up the lights for performances, but some don't. Darkness results in videos that are practically unwatchable. If you can, speak to the organizers in advance to ensure that they'll turn up the lights at the right time. Some of us have the capability to enhance dark videos in editing, but most people don't. Your goal as a videographer is to ensure that you shoot the best quality footage to begin with, not something filled with problems that will take many hours to fix in post-production.
7. For heaven's sake, don't chew gum! The camera's microphone will pick up the sound.