I don't see the app or the App Store on my Apple TV device.

If you have an older model then it doesn't run apps, and thus doesn't have access to the app store. If it's a 3rd gen then you would need to update the firmware to 7.4 so it can receive the server version of the TV app. If you have a 2nd gen then it won't be available.

Identify your Apple TV model - Apple Support

What model of Apple TV do you have exactly? See here: Identify your Apple TV model - Apple Support

You need a 3rd Gen Apple TV or later. The App is the black icon with the apple logo and the letters TV on it.

If you have the blue tv screen on the black icon instead, then you need to update the software on your Apple TV. Settings->System->Software Update.