To reach Bhutan you can get a direct flight to Paro(a valley town in Bhutan), or you can reach Bagdogra via flight or NJP station by train and get a cab/taxi from there to Phuentsholing(border town of Bhutan). Jaigaon is on the India side of India-Bhutan border, where as Phuentsholing is on the Bhutan side. You can easily see the cultural difference in India and Bhutan as soon as you cross this border. On one side people are so content and happy, on the other side you will witness chaos and population bloat up. It is really easy for Indians to travel in Bhutan since no Visa is required for Indian citizens and also for citizens of Bangladesh and Maldives. All you need is a permit which you can easily get from the Embassy in Phuentsholing.
Thimphu : This is the current capital of Bhutan(Punakha was ancient) and offer dreamy landscapes which are feasts for your eyes. The journey from Phuentsholing to Thimphu takes roughly 6hrs via taxi. If you will start your journey in afternoon from Phuentsholing, you will reach Thimphu by evening, after that you can checkin in your hotel, rest for the day, and then go for sight seeing next morning. This capital gives a mix of traditional as well as urban feeling since it has Buddhist sights and attractions, and also offers nightclubs, bars and cafes.
Places to visit in Thimphu are :
1. National Memorial Chorten
2. Trashi Chho Dzong
3. Changangkha Lhakhang
4. Buddha Dordenma
5. Motithang Takin Preserve
6. Handicraft market
You should also try the local Bhutanese cuisine when in Thimphu. I tried Saag Datshi(leafy vegetables cooked in Yak cheese) and it was really tasty and filling.
Punakha and Dochula Pass : You can have a one day trip from Thimphu to Punakha, and in between you can visit Dochula Pass also. It takes 45min to reach Dochula Pass from Thimphu via taxi and another 2 hour to reach from Dochula Pass to Punakha.