It seems harder than it ought to be to get quality audio while having an equally high quality video feed - especially trying to do it live and with more than one person involved (in different locations). What do you recommend for accomplishing this in the best possible way?
Alot of money. Audio and video are just like any other product out there, the more money you spend, the higher quality you can get. Of course, at that point, you should be somewhat of an expert in the field so you can utilize both properly.
You say feed, so I'm guessing that you're streaming these? If so, your real limit is going to be bandwidth and network capacity. If it is having to strain to manage the audio and video, or compress it to manage the data then the quality will suffer. Get a decent bandwidth and processor to stream this. No setup can give decent output if it is getting low quality input.
I didn't think about the bandwidth issue. I was trying to do it on Facebook, which (I'm sure) has its own issues with streaming and limits - as well as compression, which I also neglected to consider. I think I need a different platform.