The places you can explore in Gangtok is:
•Nathu La pass (must carry camphor, because price there is a bit high)
•Baba mandir •Chhangu Lake •Hanuman Tok •Ganesh Tok • There are numerous monastries as well ( All are peaceful and amazing) •Bakthang Falls •You must visit Pelling as well •The north part of Sikkim is also very beautiful which includes : GURUDONGMAR LAKE, ZERO POINT, NAGA FALLS, YUMTHANG VALLEY, LACHUNG
You must note that the prices of the food in Gangtok are quite expensive.
To my knowledge no personal cars or bikes are allowed in the route of Nathula.. Till Gangtok you can go and get to know more from your respective hotel, they can guide you accordingly