Besides the service.StartType = ServiceStartMode.Automatic my service does not start after installation

Solution

Inserted this code on my ProjectInstaller

protected override void OnAfterInstall(System.Collections.IDictionary savedState)
{
    base.OnAfterInstall(savedState);
    using (var serviceController = new ServiceController(this.serviceInstaller1.ServiceName, Environment.MachineName))
        serviceController.Start();
}

Thanks to ScottTx and Francis B.