I did a trip to Darjeeling last year. I started from Kolkata during the night by train, got down at Jalpaiguri.
From there, try for shared taxi; it will not cost you much.
After reaching Darjeeling, there are many hostels in the city. I stayed in International Youth Hostel. It was nice.
Apart from this, have food in MG Road which will be cheap as well.
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