The nearest airport to Darjeeling is bagdogra while the NJP is the nearest railway station. You will get a sharing taxi for 150 bucks right in front of the main gate of the railway station or you can take a rikshaw to Darjeeling more and take a sharing taxi. If you want to travel to kalimpong, you can go to panitanki and take a shared taxi for 130 bucks. Homestays are more popular in those areas than hotels.Hope it helps.
If you start your journey from Kolkata or surrounding Kolkata, then train will be the best option..
you can book your rooms through MMT or OYO, or you can go Tin Chulya ( A small village on the way to Gangtok from Darjeeling). you will get cheap home stays over there..