The Dalai Lama temple has the beautifully informative and educative Tibetan Museum, Namgyal Book shop, temple and Dalai Lama's residence. It's a beautiful place of zen & peace and lots to learn about their history and religion.
During teachings there, expect several thousand people packed in. But at the end of the day at the temple, less tour groups and people are there and it's the most peaceful time. You can go to the "teaching temple" and sit off to the side and find some moments to meditate. The energy of this place is palpable and if you are interested in spiritual places, this is a great place.