The first time in India is always exciting and also a bit exhausting. I think it's nice to get the Sikkim permit at your embassy when you get your visum for India.
If you have the money for the flight to Bagdogra, it would recommend that you treat yourself with the flight, because the train ride takes long and you're not accustomed to the food yet etc. Besides, if you take a direct
flight from Delhi to Bagdogra (not that other route where you have a stop in Guwahati) you will pass by the peak of mount Everest, which is a really majestic view.
I've heard that the national airport in Delhi is now directly at the international airport, so that you don't need to leave the airport if you continue with a flight to Bagdogra. If you arrive at Delhi at three in the morning, you will to have spend some time waiting in the line for the passport formalities. That will take at least one hour.
Then there is a State Bank of India directly at the airport, where they have very good exchange rates. (It's not
like at airports in Europe, where the exchange rates they give you are more or less a scam.) Actually you should exchange a great part of your cash there, because going to a bank in India is always exhausting, and you won't find better exchange rates anyway.
Collecting your luggage, passport and visum control, exchanging money at the bank - that will take you between about one and two hours. When you have a flight the next morning it will probably not make sense to drive to the Manjukatilla, because that is a taxi ride of almost an hour.
I suggest that either you spend the night at the airport, or you take some two or three days time in Delhi to arrive. But the question is also when are you going? Last winter they had a hell of air pollution in Delhi, these are some photographs from last November:http://www.tagesschau.de/ausland/delhi-smog104~magnifier_mtb-1_pos-1.html#colsStructurehttp://timesofindia.indiatimes.com/city/delhi/Delhi-enveloped-in-smog-back-to-pre-CNG-levels/articleshow/17070829.cms?intenttarget=no
If I were you, I would totally avoid Indian cities during the winter because of the smog. If I have the opportunity to travel to Sikkim next winter, I will stay at the airport (which is btw climatized and air conditioned) and jump into the airplane the next morning to get out of Delhi as soon as I can. From Bagdogra you can take a taxi to Siliguri and recover from the flight for a day or two and then continue to Gangtok.