I was thinking of getting another monitor the same size (24 inch) as the one I'm using now, with the intention of playing games off of it, so I can stream off of it, but I'm wondering if I'd still need OBS up, or Chrome with Twitch on it, or a third party software to view chat. How would I lay it all out?

It's entirely up to personal preference. I like one screen for the game and splitting the other screen in half for OBS and LiveSplit.

Do you have anything to look at the chat?

I stream rarely, but I can use Twitch on my phone and set that to the side.