To keep track of shows you are currently watching you can tap on the "Episodes" tab. Here you'll find your Watch List, which has the next available episode of the shows you are currently watching listed for easy reference.
To mark one of these episodes as "Watched", just swipe right on that episode.
If you want to watch the episode later, or if you are no longer watching the show, just swipe left to get the "Remove" menu.
You can also pull the list down to view your Watch History.
Finally, you can see what you're watching from your profile. Go to your profile and select "See All".
Here's where you can find shows you finished, are up-to-date on, upcoming episodes and the dates they are airing, as well as your Watch Later and Stopped Watching lists.