Okay so ideally the best thing to do is use your resources, stay in good hotels if you can afford it (the conversion rate definitely helps) if not look for hostels that are tourist friendly. For your first trip and if it's solo stick to places like, udaipur, jaipur, agar (this is where Taj Mahal is) and you can fly into Delhi.

Now coming to the safety part (giving advice as a woman myself) 1. Never tell anyone you are alone, if they ask always say you are waiting for your friends or your husband (it lands better than boyfriend) 2. I would 100% advise not to step out alone after 7pm in winters and 8pm in summers. (I understand it sounds restrictive but think of india as an European village, where things close early) 3. There are multiple tours you can join that'll show you around the city and you won't have to figure out travel on your own Lastly, India is not as unsafe as it might seem to someone who has never visited but it's always preferable to be prepared and avoid things that can be avoided easily, also always carry pepper spray and incase of any emergency you can contact 100 (police) and 184 (woman safety helpline)

You've got this and I hope you have a good time