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..
There are shared sumo runs from NJP, it will be cheaper than booked ones. Don't get those package they could be costly and they have hidden charges. Its better you explore as per your convenience.
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.
Take shared cabs from NJP (350 per person) which drops you at Darjeeling and stay at Hideout Backpackers Hostel. Connect with fellow travelers there for renting cabs to Tiger Hill, Lamahatta etc.