There really is no right or wrong here. I would go for one activity and adapter per set of data. This will make your activities less complex and make them decoupled from each other. In your case I would have:
- ShowActivity+ShowAdapter
- SeasonActivity+SeasonAdapter
- EpisodeActivity+EpisodeAdapter
Your data model (with Shows, Seasons and Episodes) should be made accessible to the activities and used to initialize each activity and adapter with the correct data.
Once a show has been selected I'd pass on the id of the show to the next activity through the intent bundle, pick up the show id in the SeasonActivity, fetch the relevant seasons from the data model and display them. The same goes for the transition from season to episode.
Using the approach of one activity per list you'll also get nice transitions between activities and don't have to worry about dealing with the back button to navigate up through your hierarchy.