Hi Megha! My hometown is near Darjeeling. It's a beautiful place. Actually it's a part of North Bengal, hence there are more places to visit. Let me jot them down for you.
Darjeeling
Kalimpong
Lava
Lolegaon
Rishop
Sandakpu
Jhalong
Lataguri
Gorumara
Jaldapara
Jiti-Bhutan (you can spend a day in Bhutan as well)
Kurseong
There are many more points which you will come across while traveling. Don't miss out to click in those places.. And all these places are must visit types.
Any further help, feel free to ping me.