There are a couple of ways to do this. First, from a podcast view, you'll see all the episodes listed. Three small dots can be found to the right of each. These dots function as a “more” button, which you can tap to find more options. From there you can add that particular episode into Likes, Listen Later, and Downloads.
Or, always from the podcast view, tap on the episode you want to listen to. Then, tap on the player at the bottom to enter the episode view. From there, tap on the heart icon to add it to your Likes list, the clock icon to add it to your Listen Later list, and the cloud icon to your Downloads list.