Best Time to visit Sikkim is October to December and March to May. Since you want to plan 10 or more days trip......below is the itinerary we followed when we visited Sikkim: Day-1:Flight to Bagdogra and reached Siliguri
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 Pedmayangtse and back to Siliguri.