There are two ways one through Train and another one is through flight.
If you are planning through train then book Mumbai to New Jalpaiguri Railway station(NJP) and if you are planning for flight nearest airport is Bagdogra Airport.
Distance from Airport or Railway station to Gangtok is almost similar that 3-4hrs.
There are various places that you can cover such as Darjeeling,Gangtok,Pelling,Kelimpong,Lachung etc
The best route is to take the flight directly to Gangtok and travel the places in Sikkim by the local sumo you get their which are generally very cheap from 150 to 300 pp to anywhere from one city to another . I chose the route from Kolkata to siliguri by train then to darjeeling by sumo then to Gangtok by sumo again. we hired the sumo to and fro from siliguri and to all the other local sightseeing which costed us 25k for 8 people. we got the cab in front of siliguri station. don't forget to bargain.
You need 5 days to weeks time for these places that can be reached by Flight upto Bagdogra Airport followed by 5 hr Drive to Reach Gangtok, Sikkim and other one is by Train and you can book upto NewJalpaiguri and then by road from there also approx 5 hr Drive. Please note you need to get Inner Line Permit for Visiting Sikkim and Special permission for Nathula Pass. Places are Lachen, Lachung, Pelling, Darjeeling, Kalimpong, etc.