Can't help you for capture card, but for streaming software, the best option is OBS Studio (https://obsproject.com/), free and easy to use; If you have trouble setting it up, or want optimized settings just search for some help online :)

For a cheap-ish capture card i would recommend Avermedia LGP lite. You cant record or stream your switch very well without one.

Im planning on just getting a phone holder and streaming on my phone from the twitch app, it probably will suck and it will be harder to find because it only has the IRL, Creative, Social eating, etc. Options to stream in

Is a laptop fine to stream from? I assume the console is doing the heavy lifting and the computer is only responsible for capturing the footage or would a lower end computer not be enough to stream it when you can stream other games that are being played on said machine?

With some research I got an Elgato Game Capture HD for $70

It's not the most amazing HD capture card out there, but it's pretty good and does the job.

@GoodleShoes Yeah, a laptop should be fine as long as you run at a lower bitrate, say 500-750. If you don't have good internet, you should also consider streaming at a lower bitrate. You can configure it in OBS' "Stream" settings window.