Re: shooting HD vid on a TIGHT budget... need clues
If you can get your stage area to be well lit, you will have better results... a dark stage setting is a nightmare in cinematography.
GoPros are fine but they have MAD barrel lens distortion which is usually corrected in post in the best GoPro videos. I've been a longtime user of their Cineform codec and I really like what GoPro does.
60fps is only really useful for slowing down footage in 30fps or for 60fps output. The best thing GoPro has is the ability to shoot in 2K resolution with a REALLY wide angle... the cropped medium and narrow settings don't quite have the resolution of the wide settings. There's no ability to use any manual settings like shutter speed, aperture and ISO (film speed) with the GoPro... this is typical of sub $500 cameras.
I've been pleased with my cheapo
Canon Powershot ELPH 330HS which shoots in full 1080 HD and has decent audio. It has some limited manual settings like shutter speed and ISO but has no aperture control. Focus is automatic so there can be some issues of blurriness or being slightly out of focus. Getting good automatic focus depends on proper exposure (which depends on lighting), centering the subject in the frame and making sure there are no objects with which the camera can focus on in the foreground
instead of the subject. Over or under exposure will throw off the automatic focus... with over exposure creating the most blur.
A wide aperture is usually called for when shooting dark scenes... but the problem with that is twofold. People usually pick the lowest (widest) aperture setting that their lens can go... this setting is usually really hard to get in crisp focus. Don't open the aperture all the way... stay one stop back from the lowest setting the lens can go. This will result in better focus. The second problem with opening the aperture to let in more light is that the depth of field increases, making precise focusing much more difficult.
You'll have to record the audio separate and sync it in post... or you could gamble and use an audio "in" jack on whatever camera you get your hands on and hope you get good levels (level monitoring is difficult on most sub $1,000 cameras).
So, the most important part is that you are well lit without any visible overexposure to the naked eye. The camera will calibrate itself perfectly in "AUTO" mode by tipping just a couple of areas into full overexposure (GOOD) which gives you dark and better resolution blacks. your camera will constantly be trying to turn the black areas into dark grey areas if there isn't at least one point it can see as pure white. A small amount of overexposure is not a problem at all and is actually desirable since crushing the blacks in post is a normal thing to do when processing cheaper cameras.
I am DEAD serious about spending more time on the lighting and audio than anything else. I have no doubt you will record stellar audio so my advice is video only.