Rajasthan is also a fantastic spot to visit if you want to see palaces and temples built by Rajput monarchs in the past. As you travel around Rajasthan, try to include as many of these great spots to visit as possible on your plan.
Udaipur
Jaisalmer
Jaipur
Jodhpur
Pushkar
Bundi
Ranthambhore National Park
Chittorgarh
Bikaner
September is a good time to visit Kashmir, at this time the autumn kicks in and you can see the splendid beauty of Kashmir. Plus Monsoon also retreats by the time. So go for it. Heaven on earth awaits you.
Public transport is available in abundant for travellers who want to travel to any spot they like. There are fixed timings and fixed stopovers. Bike travel is our own mode of travelling where we can travel at our own pace. I would say bike travel is the best.
For clear, fresh air and sunny skies- October to December is the ideal time. January and February are colder, but from then until April the climate remains dry and pleasant and in late spring you can witness famous blooms in the valleys. I hope this helps you. Keep exploring!