ok, so the select / from portion of your query:
SELECT featured_shows.feature_id, featured_shows.event_id, featured_shows.feat_title, calendar_event.id, calendar_event.date_start
FROM featured_shows, calendar_event
needs to be changed to use inner join:
SELECT featured_shows.feature_id, featured_shows.event_id, featured_shows.feat_title, calendar_event.id, calendar_event.date_start
FROM featured_shows
INNER JOIN calendar_event
ON featured_shows.event_id = calendar_event.id