Try staying in hostels..... Best Time to visit Sikkim is October to December and March to May. Below is the itinerary we followed when we visited Sikkim:
Day-1:Reached Siliguri by evening
Day-2:Siliguri to Gangtok and applied for necessary permits.Sight-seeing in Gangtok.
Day-3:Gangtok to Lachen.
Day-4:Gurudongmar Lake and Kaala Patthar. Reached Lachung by evening.
Day-5:Yumthang Valley and Zero Point.Back to Gangtok.
Day-6:Nathula Pass and Tsomgo Lake.
Day-7:Char Dham in Namchi. Reached Ravangla and visited Buddha Park.
Day-8:Rabdenste Ruins and back to Siliguri.
Hello , 10 days for traveling to Sikkim is an awesome idea.
You can take flight till Bagdogra and then there is the Cab service known as Wizzride ,you can book your seat directly from Bagdogra Airport to your hotel Room.It costs around 800 per seat.
Its a spring season but you may witness lots of rain in the evening .
Rhododendrons are now blooming in the West of Sikkim must visit place known as Barshey. In 10 days you can explore North Sikkim ,which includes Lachen ,Lachung ,Gurudongmar Lake.
There is a skywalk in Pelling.You can visit South Sikkim for visiting tourist spots.
There are many hangout places in the nearby M.G Road If you are a food explorer you can try "Laphing " and authentic Nepali food known as "Thakali".
I hope you have a very adventurous and life's best trip.