How to automatically start your service after install?

Add the following class to your project.

using System.ServiceProcess;  

class ServInstaller : ServiceInstaller
{
    protected override void OnCommitted(System.Collections.IDictionary savedState)
    {
        ServiceController sc = new ServiceController("YourServiceNameGoesHere");
        sc.Start();
    }
}

The Setup Project will pick up the class and run your service after the installer finishes.

Small addition to accepted answer:

You can also fetch the service name like this - avoiding any problems if service name is changed in the future:

protected override void OnCommitted(System.Collections.IDictionary savedState)
{
    new ServiceController(serviceInstaller1.ServiceName).Start();
}

(Every Installer has a ServiceProcessInstaller and a ServiceInstaller. Here the ServiceInstaller is called serviceInstaller1.)