As per my experience to Sikkim, I would recommend these must to do things if you are an explorer and adventurous. Homestay in North Sikkim Interact with monks in the monastery and experience their lifestyle for a day River rafting in Teesta river Yak ride in north Sikkim Good Luck!!!
Happy Travelling!! Hot water spring bath near Bumthang Valley If possible interact with trans border Himalayan road workers.
For Darjeeling
Don't miss the toy train ride
Batasia loop
sun rise at tiger hill
Himalayan Mountaineering institution
zoo
mall
kalingpong
japani temple for sikkkm
gangtok
baba mandir
Tsongu lake
lachung
peling
and many more. can help you if you can precisely say North or south
North sikkim is great...
Laghung
Nathula pass Darjeeling
Kalimpong is a must
toy train
Himalayan Mountaineering Institute
Padmaja Naidu Zoo
Tiger point for sunrise
Japanese temple
Shanti stupa
Tenzing rock
Kurseong for sunset
Tea garden
North sikkim is great...
Laghung
Nathula pass Darjeeling
Kalimpong is a must
toy train
Himalayan Mountaineering Institute
Padmaja Naidu Zoo
Tiger point for sunrise
Japanese temple
Shanti stupa
Tenzing rock
Kurseong for sunset
Tea garden
In 6 days trip, you can cover Darjeeling in a day from morning 4 at tiger hill to batasia loop, toy train ride from ghoom to Darjeeling and then Darjeeling's local sightings of hmi, zoo, happy valley, rock garden, chowrasta. next day you can go to kalimpong and cover the local sightings. 3rd day you can leave for Gangtok and take a day for local sightings and the next day you can cover nathu la pass, tsongmo lake.
you can have a look at my blog prepossessing hills which has a complete details of the places in Sikkim. it will be very helpful.