I can not find an API method method that would provide me with a set of favorite/ignored tags for a given user. Is there such a method? If not, is there a recommended way to get this information?
No, the API does not provide a method to get the user's question-page, display preferences -- including Favorite or Ignored tags.
Currently, the only way for your app to get these is to browse to the questions list and scrape the tags from the right-hand column.
If your app is running in the user's browser and the user is logged-in, this is a piece of cake.
Otherwise, your app must somehow obtain the user's login credentials, which most users would be justifiably unwilling to provide.