Hi Hari,other than rainy season people do visit all the time but for best scenic beauty it's from March to May. Budget- depends on how many days u plan
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March, April, October, November are the best months for Sikkim. If you want to spend your weekend in the midst of ONLY Tea garden in Sikkim, you can visit this link :
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March and April.....Budget will be around 15-20k
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Best time is March-May. But the bestest is April. But you can also visit in May. It will be best this time as there was a lot of snowfall so you'll find a lot of it at Tsomgo Lake and other places. Budget depends on number of places and days. For a complete tour of like 9-10 days it will be like 20-25k/person. You can reduce it by staying in homestays or camps or something .
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As suggested by Sikkim Tourism, the best time to visit Sikkim is either between March and May or October and mid-December. If you want to witness the blooming natural beauty in Gangtok best season to visit would be in spring, from March to May. your budget must be more than 10K