Best cities could be a matter of debate as it is subjective. What is it that you are looking for? Where are you located and what are your preferences? South? North? Eastern part? Please elaborate, what is it that you want from the destination so members can suggest something.
There are grades in the kind of hotels and travel one can choose from. When a single lady travels it is safer to hitch on to government run hotels and travel choices. Your name will be recorded when and wherever you travel. The Yatri Niwas group of hotels are scattered over the length and breadth of India. These hotels are cheap and safe. A lady that travels alone must always take down the number of the vehicle that she travels and also take a photo of the driver. Always carry two sims. SMS these details to the hotel reception If details of your interests are known a travel plan could be outlined for you
Females are free to travel all over India. It really depends what your interests are and which place you like. eg Hill stations,beaches, monuments etc.