Basically, ES6 is just ES5 with a bunch of new features. ES5 is not outdated as many browsers still do not fully support ES6 specs (Node.JS LTS doesn't either). But I guess, yeah, you should start embracing ES6, now, in order to write new Node.JS APIs.
Here is a list with new features and a comparison to ES5. You do not have to enforce everything at once. Just read through all new features and then use the things you can remember. Once you feel safe using them, take a look at other ES6 features which you can start using.
rollup.js to bundle everything into single file. The only one thing in ES6 I would avoid is classes and
new keyword. That will make everything even simplier.
By using TS over JS you will loose simplicity and flexibility and, probably, will have a Gorilla/banana problem with a lot of interfaces and classes.
The other answers already speak about ES6 and I agree with them there. I will restrict my answer to TypeScript.
You should definitely be using ES6 for any new code where possible. It's really easy to pick up. That's the way forward and everyone is using it. There's really no downside to using it, except that there's an extra build step to convert it to ES5 but you should really already have a build step for your project.
As for TypeScript, definitely give it a try after moving to ES6. Chances are you'll like it if you give it a shot :)