Let's say I have a deployment template like this

spec:
  minReadySeconds: 15
  readinessProbe:
    failureThreshold: 3
    httpGet:
      path: /
      port: 80
      scheme: HTTP
    initialDelaySeconds: 20
    periodSeconds: 20
    successThreshold: 1
    timeoutSeconds: 5

How will this affect the newly versions of my app? Will the minReadySeconds and initialDelaySeconds count at the same time? Will the initialDelaySeconds come first then minReadySeconds?

From Kubernetes Deployment documentation:

.spec.minReadySeconds is an optional field that specifies the minimum number of seconds for which a newly created Pod should be ready without any of its containers crashing, for it to be considered available. This defaults to 0 (the Pod will be considered available as soon as it is ready). To learn more about when a Pod is considered ready, see Container Probes

So your newly created app pod have to be ready for .spec.minReadySeconds seconds to be considered as available.

initialDelaySeconds: Number of seconds after the container has started before liveness or readiness probes are initiated.

So initialDelaySeconds comes before minReadySeconds.

Lets say, container in the pod has started at t seconds. Readiness probe will be initiated at t+initialDelaySeconds seconds. Assume Pod become ready at t1 seconds(t1 > t+initialDelaySeconds). So this pod will be available after t1+minReadySeconds seconds.