Here is my live stream setup kit :)
If you are doing a personal stream, I recommend only streaming to a single service such as Twitch. It is better for community building and easier on the streamer. Additionally, if you become an affiliate you are locked into a platform anyways.
How my setup works is that I have my main Desktop PC that has my streaming software, chat, alerts, music, and such all running on it. I have a second surface book that has HDMI out into the capture card in my Desktop PC. I do this because often when you compile apps it may freeze up your computer and stream :(. I have 2 sets of keyboards and mice and 2 monitors that I work off of. This works for me, but you do you :)