Reflectors and diffusers (+ most likely lights for the indoor shots).
People holding said reflectors and diffusers
Lens with good focusing ring and image stabilization capability (sharpness and high low light performance are a plus).
Boom mic or button mics for the sound - possibly external recorder for phantom power or for wireless audio.
Tripod (and maybe light stands).
Editing software capable of masking and motion tracking (like AE).
Model release forms for the people appearing in the background.
That is, if you want to make it look somewhat professional with not trying to squeeze blood from a stone. Youtube channel Film Riot offers very good tutorials how to do things on the cheap while maintaining quality.
A gopro is the most versatile camera on the market right now. They are affordable, get phenomenal picture quality, have a wide-angle lens for group shots, can be used for more than just live-streaming, and there is a plethora of attachments that can be used to make your life easier.
As for sound quality for the dollar, you would be amazed at the Blue Yeti microphone. It's a versatile microphone that can be set to unidirectional (if you are by yourself), bidirectional (sitting across a table from someone), or omnidirectional (sitting on a couch with a group of people) recording modes. You can plug your headset into it when your are recording and it supports plug and play over USB. The normal version costs $100, but if you spend an extra $80 you can get the pro version. The pro version also supports 24-bit audio and an XLR connector if you want to invest in a pre-amp down the line.
My opinion on mixers is that most people do not need them. They are great if you have high-end condenser microphones that require an external power source. but that is such specialized technology that it would be better to get your podcast up and running first.
Edit: The Wirecutter recommends the Blue Yeti for a microphone as well. In case you did not know, the Wirecutter is a site that is dedicated to showing you what the internet consensus is for purchasing decisions. It should be the first place you look when you are trying to find what item to buy.
Really good recommendations. GoPro should be just few bucks more expensive than decent webcam but way more versatile.