You can't compile your code automatically like you did before with the first version of the Eclipse plugin. The new plugin makes use of the Language Server Protocole which will help you find errors locally, but you'll still have to use sfdx force:source:push to compile everything.
If you have access to the Trailblazer Community, there is a thread about this here.
Additionally, you can automatically push your changes to your Scratch Org using Nodemon, as described in this blog post.