Well talking particularly about gangtok city, there is not much to do in it. There are few sightseeing points like ganesh tok, tashi view point, some monastries, and few others but these are not the places for which people visit to north east.
If u really want to see something different, then visit the changu lake, baba mandir and nathula pass which are approx 50km from gangtok. U need a take a permit to visit these places which are issued same day of your visit but you have to submit your documents 1 day before your visit to your respective driver/travel agency.
Enjoy the glacial Tsomgo Lake which is s located at an altitude of 3,780 meters above sea level. It is also called as Changu Lake and is crossed by the Nathu la Pass on the north. Its a beauty catch to your eye. During winters this lake is at its best. Ahead on the route theres also a tourist attractions Baba mandir. During heavy snowfall this route to Baba mandir is closed usually. Later you can visit Hanuman tok and in evening enjoy the walk to MG Road.