If you are persisting data (such as podcast urls, episode information, etc.) then you should store all of it in one place. Core Data can do that easily.
How much data are you storing? I would guess from your app description that you have more data than just what is a favorite.
Update
If you are storing a favorite flag you need something to associate it with. A URL perhaps.
If you are storing the URL, you might as well cache the podcast name with it.
If you are storing the name, might as well cache the episodes.
See where I am going? :)
Keep a local cache of what you are getting from the web and the user gets a better experience because they see data on launch, they don't need to wait for the internet which is random.
If you are caching data, then yes! use Core Data. Otherwise, NSUserDefaults
is probably sufficient.