The famous Tigers Nest monastery (on most tour brochures) while cliche, is actually a phenomenal experience. I'd suggest you arrive mid-afternoon as the evening light makes for a truly breathtaking experience. Just take a torch so you can get can get back down in the dark.
Actually, if you check my Flickr stream you'll see some photos from when I was there last year and one tip is ALWAYS aim to be at temples and monasteries late afternoon as they light prayer candles and the like as the sun disappears.
My experience in Bhutan was probably the best of any travel destination I've ever been to. Priceless.
Have fun! almost 8 years ago