The usual approach is by flight to Bagdogra Airport (IXB), about 90 km south of Darjeeling in the plains. From Bagdogra, shared taxis and private vehicles run up the hill to Darjeeling in 3-4 hours depending on traffic and road conditions. Several Indian cities have direct flights to Bagdogra. An alternative is flying to Kolkata and catching an overnight train to New Jalpaiguri (NJP), the main rail junction for the Darjeeling hills.
From New Jalpaiguri, the Darjeeling Himalayan Railway runs its famous narrow-gauge service up to Darjeeling in 7-8 hours — beautiful but slow. Most travellers take a shared taxi instead, which takes 3-4 hours and costs a fraction of the train fare. The NH10 road climbs steeply through Siliguri and then winds through the tea gardens.
The toy train between Darjeeling and NJP is worth doing once for the experience, but expect delays. The steam service only operates on the upper section between Darjeeling and Ghoom.