Do visit nathula pass, tsomgo lake, baba mandir, lachen, lachung, gurudongmar lake, pelling.. Sikkim is vast and would require atleast 13 days to explore. Do visit the gangtok market. there are laods of eating joints there like Roll House, Dragon walk, thakali, cafe live and loud, Bakers street etc!!
The best things to experience in Sikkim are : visit lake tsomgo , rumtek monastery ,gurudongmar lake , goecha la , rafting in teesta river , ropeway in gangtok , paragliding in gangtok .