Over the last seven years, I have seen them turn as early as the second week of September, and as late as the last week of September. It all seems to depend on the weather. As for those bugling elk, I am quite partial to the early morning hours. As the temperatures rise during the day, the level of activity seems to decrease. This doesn't mean that you won't see and hear them during the day, it just means that there is more going on during the first daylight hours.
Evening is also a really good bet.
And there are so many hiking trails to choose from, it really depends on what you would like to see - wildlife, scenery, etc. If this is your first trip to the park,then I would probably recommend several shorter hikes to specific landmarks or sites such as Fairy Falls or Lonestar Geyser.