The closest Airport to Sikkim is at Bagdogra, 130 kms from Gangtok (the capital city of Sikkim), where direct flights operates from Indian cities like Kolkata, Delhi, Guwahati, Mumbai, Chennai, Paro and Bankok, and all Indian major cities via different cities, travel time to Gangtok from the airport is approx 4 hrs.
One can also travel from Kathmandu to Bhadrapur (Nearest Nepalese Airport to Sikkim), in the eastern Nepal (approx 1.30 hrs, from Siliguri (Gateway to Sikkim-Darjeeling), via Kakarbhitta (Internationl boder between India- Nepal) and then drive to Gangtok (114 kms, approx 4 hrs).
The nearest railway station from Sikkim is New Jalpaiguri which is connected to Kolkata, Guwahati, New Delhi, Banglore, Chennai, Mumbai, Jammu, Etc. and other major Indian cities.
Gangtok is connected to nearest town like Siliguri (114 Kms, 4 Hrs), Darjeeling (97 Kms, 4 Hrs), Kalimpong (80 Kms, 3 Hrs), Phuntsholing (Bhutan) (220 Kms, 6 Hrs).
In addition to an Indian visa, foreigners and NRIs must obtain Inner Line Permit (ILP) to visit Sikkim, the permits can be obtained from all Indian missions abroad, Ministry of Home Affairs (MHA), New Delhi, Sikkim Tourism office, New Delhi, Sikkim Tourism Office, Kolkata and Sikkim Tourism Office, Siliguri and in Arrival at Rangpo/Melli (boder to Sikkim) on the strength of an Indian Visa. Foreigners and NRIs are issued a permit for initial period of 15 days duration on the spot without any delay provided photocopies of Passport and Visa details along with two passport photos of applicants are made available then and there. It can be extended for further 30 days from the Home Dept., Govt. of Sikkim, depending on the merits of the case. If you are visiting the interior regions, you also require a Protected Area Permit which is obtainable in Gangtok from the Department of Tourism. Utkarsh Travels will handle all the formalities for you to visit.
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