For Gangtok:
Reach Siliguri and either take a taxi or bus to reach Gangtok.
It will take around 6-7 hours to reach to Gangtok. So you will not have much option to visit on the first day of arrival.
Although you can walk around the city and MG road. Day2: Go to any tour operator and book a one day tour to Nathu La Pass, Changu Lake, Baba Harbhajan Singh Mandir. (Negotiate hard. For shared taxi I negotiated for 600 per person)
Day 3: If you have more days to spend in Sikkim, then you can opt for North Sikkim tour which include 2N 3D trip to Lachen(Gurudongmar Lake, Kala Patthar will be covered) and Lachung(Yumthang valley, zero point, flower valley etc). I will suggest you to opt for this. You have to again book it through your operator and don't forget to negotiate with them. They usually charge high.
(For shared taxi, I paid 2800 which included transfer, accommodation and food).
After you return from North Sikkim you can go for Gangtok city tour. There are many beautiful view points, waterfalls and monestaries.
You can spend less days in Darjeeling but spend more days and cover all places in Sikkim. It's really very beautiful state.
I hope this will be helpful for you.
Visit Tiger Hill to eyewitness a beautiful sunrise
A memorable ride Darjeeling Himalayan Railway (Toy Train), a UNESCO World Heritage Site
Visit Directorate of Handicraft & Handloom, White Hall & Flower Show
Enjoy a fun cable car ride in Darjeeling and Gangtok
Yak ride at Tsomgo Lake and Visit Baba Mandir
Explore Rumtek Monastery, Do-drul Chorten and Research Institute of Tibetology.
Visit to Tea Gardens, Batasia Loop. Ghoom Monastery, Peace pagoda,and Japanese Temple