For sight seeing in Darjeeling refer here https://www.indiatravelforum.in/threads/two-days-in-darjeeling.206/#post-560
you could get yourself a taxi from Darjeeling to Gangtok which is the most convenient way or else get a bus which runs from Siliguri Bus terminal to Gangtok bus stand..
Gangtok has many beautiful places According to me its better than Darjeeling.. i don't exactly remember all the places i visited but the ones which i remember are :
#Teesta river that you would see on your way to Gangtok, even rafting is being done here.
#Rumtek Monestry : it has a complete calmness ,relaxes mind and soul.
#Ganesh Toc point: it s lord Ganesha temple situated at some great height from where you can have a look at the entire city
#Nathu la pass : Sikkim-tibet border its a nice place.. we missed the chance to visit this place as it was nearly 50kms from Gangtok. and we had no time...
#Banjharika falls : its a different type of fall, it was falling from mountains by the side of the road...
#Cable car : have time so get in it and enjoy the ride
#Ridge park : there a flower exhibition a variety of flower await u there
All these are worth watching
