Catch a train from Kolkata to New Jalpaiguri, the gateway to Northeast India and Sikkim. The journey takes approximately 10-12 hours. Book an overnight train if preferable.
Take an early morning shared jeep or hire a private cab to Kathmandu, Nepal, via Siliguri and the border at Panitanki (India) and Kakarbhitta (Nepal). The journey will take about 10-12 hours including border formalities.
Travel by road through Phuentsholing and border crossing back to India at Jaigaon, then onward to Gangtok which can take around 8-10 hours including border formalities.