we can help with details here rather than searching...
Camcorder:
Typically, you will see the "professionals" using "big bulky" camcorders because they have large diameter lenses to allow in light (more than itty bitty lenses) and they have a large imaging chip (or 3-chip array) to deal with the light coming in through the lens. That they are large allows them to deal much better with poor lighting situations. Any camcorder can record decent video under good lighting conditions. The wider window of opportunity from the large lens and large imaging chip means use/purchase of that single large bulky camcorder rather than investing in different camcorders for different lighting conditions.
Audio:
Since you never know what the audio infrastructure will be in each venue, it is difficult to know what might be available from the house. Even if there is an audio board/mixer, there may be bad channels, outputs you want might be used for something else, or the venue or other performers may not want their performance recorded. If all it is is standup - no instruments, no other audio cues involved, then it is pretty straightforward - single wireless lave on the talent. Done. As soon as you stray from that, things can get complicated pretty quickly. Lots of variables. Wireless mics are handy - and those which will not have as much of an issue with interference from power cable, lights (stage and otherwise), and all matter of radio frequency and electromagnetic frequency interference) will not be inexpensive. You can try the low end stuff - but again, look at what the pros use. While they would rather have more money in their pocket, they buy what they buy because the gear is more rugged and chances for picking up interference is a lot less.
An alternative is to record the audio using an external recorder and get the house audio from the house mains speakers (if they are being used). Be sure to place the audio recorder near the house speakers or whatever the audio output is. This, like an external mic, decouples the audio from the framing of the video. That way, you have little dependency on the location of the of the internal camcorder mics.
Generally, it will be easiest to place the camcorder at the back of the house and zoomed in on the stage. With any luck, the lighting is not as poor as you think it is on stage as compared to the audience "poor lighting".
For example:
https://www.youtube.com/watch?v=obYN21ZQtco
The house audio was recorded by the camcorders. I brought lighting in (two light trees, each with two PAR56 cans). The camcorders used were a Sony HDR-FX1 and HDR-HC1. I know this video goes back a couple of years, but the principles remain the same.
If you budget is for the camcorder only, you are in the high end of consumer camcorders. The Canon HF G25 is a good one - as is the Sony HDR-CX900 and you may be able to add other gear over time. If your budget includes tripod, wireless mic, case and other stuff, I don't think we can get there from here as the wireless mic (Sennheiser or Shure UHF lavalier) may consume too much. An external audio recorder might work (like a Zoom H1).
If pressed, a shopping list:
Canon HF R500 (I don't know if there is a PAL counterpart) - be sure whatever camcorder you select has an audio input. In your budget range, this will be a 3.5mm stereo audio input. Many low end camcorders have no audio input at all, so be sure the specs have that listed. You'll want 1-2 optional high capacity batteries in case the camcorder needs to be placed where there is not power available. The battery in the box with the camcorder likely won't be enough.
Shure Sony UWP-D11 (I use Sennheiser G3 labs, but Shure's good, too).
A fluid head tripod - not fluid-like - Magnus and Davis & Sanford have "low end" ones (less than $150).
Pelican cases are my choice.
We probably blew through your budget. Expect the small lens and imaging chip on the R500 to be a problem under poor lighting conditions.
Lighting: up to you. Light trees are cumbersome. So are PAR cans and their travel cases. Getting to the venue 2 hours ahead of everyone else and setting up, extension power cables, etc... and being the last to leave after packing up can be tiring. You may not be allowed to pack up until all the customers are gone.
"Easily edit and upload"... Define "easy".
The first time I use PowerPoint, I had no idea what I was doing. But I learned. It is now easy.
The first time I sat in front of a video editor, I had no clue. There is no other interface like it. Video, especially high definition video, is a computer resource hog. It will use every CPU cycle and available hard drive space. You will want to edit, add titles, insert transitions, add credits, and render. We don't know what computer you want to edit on, or what you plan to edit with. I started with iMovie on Apple Macintoshes and moved to Final Cut. The equivalent in the Windows path would be MovieMaker, then to Sony Vegas or Adobe Premier. Any high definition video will have challenges - whether AVCHD compressed (like the Canon HF R500 and most other consumer camcorders) or MP4 (the manufacturers are moving to this) or even MOV files with h.264 format video in them. ALL require the editing computer to have a fast CPU, lots of RAM and (especially in your case) a ton of external hard drive storage space. After editing for a lot of years, I can say it is "easy" but I don't do half the stuff the video editors I use are capable of doing... and this is a hobby for me.
And you will probably want (and need) the other performers to sign releases (there are boiler plates available on line). And you'll probably need to get permission from each of the venues (preferably in writing) each time you record.