The problem with video is that it is not a what you see is what you get tool. Our eyes see very differently than current video cameras. We can see, say 1000:1 in contrast. A good video camera is around say, 250:1. Reducing contrast by filling in shadows and dimming hot spots is the only way to make video look good. Color is also an issue. Where you see sunlight as white and the lamp in your house as white light, the camera sees them as shades of blues and reds relatively.

Video taping on a stage such as your performances may be problematic to say the least. Stage lights specifically are designed to have serious hot spots and high contrast. All of that said, when you are taping you will want to request some changes in the lighting to help your video. Add a camera operator who is quick with an iris and you can do it. For sketches, you can control the light and you will be fine. You connect a video monitor to the camera so that you can see what is being recorded and then you can adjust the lights to what you like.

For the camera, there are tones of choices out there. If these tapes are going to be for more than just your fun, audio is a major consideration. In that price range, good audio as well as a good camera is hard to find. There are only two that come to mind that I like in that range for semi-pro style stuff. That is Sony's HVR-A1U which runs about $2,000 or a little more with rebates. The Canon GL-2 runs about $1,800 - $2,000 but you have to buy the MA-300 audio adapter for it. I personally prefer the A1U, but I have a GL-2 as well.

Remember to budget for a good tripod and head. $300-$700. A video tripod head is much more expensive and much more critical than one for still photography.

Hopefully other will have suggestions so that we can all discuss pro's and con's with you. I do not believe that there is just one "best" tool. It is what fits your needs.