How can i get my container to go from starting -> healthy

My specific question is if I can control my container so that it shows'starting' until the setup is ready and that the health check cansomehow be started immediately after that?

I don't think that it is possible with just K8s or Docker.
Containers are not designed to communicate with Docker Daemon or Kubernetes to tell that its internal setup is done.
If the application takes a time to setup you could play with readiness and liveness probe options of Kubernetes.

You may indeed configure readynessProbe to perform the initial check after a specific delay.
For example to specify 120 seconds as initial delay :

readinessProbe:      tcpSocket:        port: 8080      initialDelaySeconds: 5      periodSeconds: 120

Same thing for livenessProbe:

livenessProbe:  httpGet:    path: /healthz    port: 8080    httpHeaders:    - name: Custom-Header      value: Awesome  initialDelaySeconds: 120  periodSeconds: 3

For Docker "alone" while not so much configurable you could make it to work with the --health-start-period parameter of the docker run sub command :

--health-start-period : Start period for the container to initializebefore starting health-retries countdown

For example you could specify an important value such as :

docker run --health-start-period=120s ...