I just want to work on my own theme (using an existing theme as a base).
I also want to, obviously, compile my own .scss files.
Whereever I search online for how-tos and tutorials I see conflicting information. Some say Slate is the new things, other tutorials use ThemeKit.
What's the correct way to move forward?
I can understand it is all a little confusing right now as both Slate and Theme Kit are under heavy development moving towards a 1.0 release upon which I am certain Shopify will do their best to provide a lot of tutorials and guidance.
Slate is an all in one, opinionaed theme development framework for Shopify. If you are in any way familiar with React, Slate is to Shopify themes what Create React App is to React apps. It comes loaded with Webpack + dev server, Babel for ES6 transpilation, a nunch of loaders like for example SASS so you don't need to worry setting that up with Webpack and most interestingly with regards to your question here: it comes with deploy environment.
Deploy environments let you specify your live Shopify environment so that Slate can then publish your theme via CLI. Under the hood, Slate does that using Theme Kit which is a CLI tool for publishing and keeping in sync theme assets with your themes on any of your Shopify stores.
So Theme Kit does the actual up / download stuff of assets, the deployment step once you're happy with an iterration on your local machine.
Slate itself uses Theme Kit for the deploy task, but also comes with a bunch of other goodies for development as noted above. You'd definetely want to use Slate as a theme developer. You'd want to use Theme Kit stand alone if you want to develop themes your way using an entirely different set of tools (say, Parcel, Typescript, or different kind of theme such as a PWA or simply because you want to).
Hope this helps!