It's a good time to go.
Try staying at Gangtok and you will get shared or private wheels with the required permits from there itself so don't go for any packages
Gangtok, lachung, yumthung valley and zero point.
Gangtok, nathula and tsomgo lake, and enjoy Gangtok with mouth watering foods and shopping.
This would be a great option if you got more time you could also visit pelling.
Ya everything there can be covered in shared cabs .. cost depends on your way of travelling for me it was only around 6k for 6days including food accommodation and taxis and the way I travel might be different from yours. Anyways it's not going to be much expensive there..
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: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 back to Siliguri.