I have attached with this comment few packages which includes all the things that are must do, you can click on them to find more details about them. If you want to talk to someone about the packages simply fill in your details in the form and you will receive a call.
I run bike tours in Bhutan, so I take myself to be well qualified to answer your question. Like most of the tourists you'd also get the permits till Punakha (Western Bhutan). Try visiting these places. Thimpu - Changangkha Lhakhang and Zangthoperi Lhakhang monasteries. Thimpu has a vivid market which deals in textile. Also, there is the historical museum if you want to know more about the history. Punakha - Cross the very famous pass called Dochu-la. Visit Punakha Dzong for its famous architecture. Paro - The famous fort Paro Dzong, Kyichu Lhakhang - one of the oldest temples in Bhutan and the Tiger's Nest monastery or Taktsang. The trek has about 700 steps to climb.
In a time span of 5-6 days, you can easily cover these places. Also check out the photos of the places that I've mentioned, they are breathtaking!
If you have lot of time visit National Museum
The Paro Dzong
The Drukgyel Dzong
Or if short time must visit Tiger Monastery-a must must see sight
Visit dochula pass to see himalayas view
Rinpung Dzong Is beautiful flag fortress
Enjoy archery game by locals in paro