Bangalore,ooty in Karnataka,munnar in kerala,kodaikanal in tamilnadu are hill station where rains can't be avoided completely but weather and place will be pleasent to visit
As far as major cities are concerned, Delhi, Bangalore and Hyderabad receive comparatively less rain. Chennai doesn't receive much rain at all during the southwest monsoon, as Tamil Nadu gets most of its rainfall from the northeast monsoon, from October to December. Kerala, Karnataka, and Andhra Pradesh also experience this monsoon, as well as heavy rainfall during the southwest monsoon.
You can visit Manali, Dharamshala, Kasol, Shimla and Kashmir in Himachal Pradesh. The best place would be The Valley of flowers in Uttrakhand to visit for departing into green world.