For apps that can connect to the Apple TV app (on Apple TV box, iPhone, or iPad), having a show added to Continue Watching (formerly Up Next) would take care of that. New episodes would appear in that row when released.
For other apps and streaming services, there is currently no such automatic system. You may have to track (check) that manually. Some popular streaming services may have opted out of previously mentioned connectability for their own reasons (e.g. Netflix).
There are (iPhone) apps that may help you keeping track [2] of what you like to watch, when it comes out, alert you if you want, etc. Not strictly related to Apple TV. E.g.:
AddToTheList app - App Store preview - Apple
Reelgood app - App Store preview - Apple (also an app on Apple TV box!)
Letterboxd app - App Store preview - Apple (also an app on Apple TV box!)
Serializd app - App Store preview - Apple
Queue app - App Store preview - Apple
Sofa Time app - App Store preview - Apple
TV Time app (often not recommended for its bugginess)