Hi Shraddha, The best time to visit North Sikkim is the months of May to September. The temperature is moderate with a maximum of 28-30 degree celcius. The temperature in South Sikkim is very cool and pleasant throughout the year because it is located at a high altitude and has a lot of vegetation around it. You can see what suits you best. Sikkim overall is good in September.
Its rainy during August, both are good and cool places. you can go Namchi,The silk route Nathula Pass, Baba Harbhajan Mandir, Hanuman Mandir (maintained and care taken by Indian Army),