Phuentsholing is a town on the Bhutanese side of border with India. The city on Indian side is Jaigaon. There is nothing much to see in Phuentsholing except get used to Bhutan, make permits for travels and board a bus!
Thimpu, the capital of Bhutan, has a youngish charm about it. Ideally this city should be your first stop and it will give you an idea of things to expect in rest of the country.
Temple of the Divine Madman at Chimi Lhakhang is a 30 minute pleasure walk thru fields and villages from the Thimpu-Punakha road. Take a look at the elaborate decorations on houses
Wangdue Dzong is one of the most beautiful monastery locations in Bhutan. It is placed on a mountain top surrounded by a large valley and thus it commands a strategic view
Cheli La (or Cheli pass) is the highest motorable road in Bhutan. Located 35 km from Paro, this point offers beautiful views of both, Paro and Haa valleys. Since there was no public transport, I hitched my way to this location :)
Thaktsang Lhakhang is one of the most famous monasteries in Bhutan. It was built around the cave where Guru Padmasambhava (Rinpoche) is said to have meditated in the 8th Century.
Trekking to Taksang Lhakhang is an amazing experience!