If you love the rains - the hills of Kerala would be a terrific place to be during those weeks(sticking to limited outdoors itinerary). Stay at some of the upscale resorts in Kerala (Alappuzha, Kottayam, Thekkady, Vythiri). Depending on how the monsoons advance, Karnataka would also be a great place those weeks (the hills of Chikmagalur) If you are not rain-lovers, then stick to reaches of the Himalayas up north - but chances are that soemtimes rains advance to the Himalayas by the fourth week - so don't take narrow or landslide prone roads.