How can I replace Mavensmate and Sublime Text with something supported?

I use these three: SFDX, the Force CLI and ForceCode (a VS Code Plugin).

SFDX can be used to work with Sandbox, Production, and Scratch Orgs.

The Force CLI is a grandparent of the sfdx plugin. SFDX represents a new era of development and is not yet optimised for all possible use cases and takes some time to ingest.

If u want to have a quick start with development / saving against an existing sandbox I strongly recommend ForceCode for VSCode. Set Password + Token and be done.

In the future, the Salesforce VSCode Plugin will surely replace ForceCode sooner or later. But for now, its a fast and easy way to work with existing orgs.

You can use SFDX with normal sandboxes, but the exact steps needed will depend on your IDE. From a CLI perspective, you need to do this:

sfdx force:project:create -n "desiredProjectName"
cd "desiredProjectName"
sfdx force:auth:web:login -s -r https://test.salesforce.com

At this point, you will log in through your browser. Afterwards, you can use force:mdapi:retrieve to get metadata, force:mdapi:deploy to deploy metadata, and the other various force: commands can be wired up to the relevant commands. For example, force:visualforce:page:create creates a new Visualforce page.

I'm not terribly familiar with VSCode, but it seems to me that one could write an extension to fully integrate with VSCode, as James Boggs is currently doing (this version only has 3 commands at the moment). All of the stuff you need to work with the CLI is basically functional, it just needs to be hooked up to the IDE.

If you want something that's basically already built for SFDX, use the Force.com IDE 2. Mind you, that means you'll have to use Eclipse (but I don't mind Eclipse), and get all the benefits of using SFDX, and none of the hassle. If you don't like Eclipse, then you're free to bring your own IDE to the table, as I said before, but you'll have to wire everything up yourself, or wait until someone else gets around to it.