Ammi Jaan kahti thi Railfanning se bada koi dharm nhi hota or koi Railfan chhota nhi hota - Shanzil Kabir

There is no rail connectivity to the state of Sikkim as of now. The nearest railhead is a place called New Jalpaiguri ( code: NJP). You can find the list of trains here /search/664/444
From NJP, its a 3-4 hr drive to Gangtok, the state's capital. You will find various options to travel from NJP to other places in Sikkim as well. However, you are advised to plan your travel in such a way that you reach NJP between 7 am to 3 pm. Before or after these times, the transport options are very limited.
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major junction is Siliguri but not many trains from Delhi stop here. Moreover, its only 10-15 kms from NJP. You can always travel to this place from NJP( rickshaw/local train, shared cab etc) and get even more frequent connections to Sikkhim.