If you're doing it strictly for learning, webrtc might not be bad to dabble with (something I want to do as well). For a real project I'd probably go w/existing solution for early iteration.

What'd you use for chat?

I would look into Jitsi (open source) and/or use JaaS (Jitsi as a service) to “cheat” and see if the api suits you. There is a generous free tier:

Web-rtc. Browsersupport.