You should definitely explore North Sikkim first as it remains one of the most underrated destinations on India . Some of the locations you shouldn't miss are :
> Tsomgo Lake
>Nathula Pass
>Lachung > Yumthang Valeey > Gurudongmar Lake If time permits you should also try to cover Pelling and Ravangla and few beautiful monasteries like the Rumtek . I think you should definitely think of a budget of 20-25k per person to explore this part as transportation is a main challenge there and tends to get a bit expensive . Hope this helps