I won't recommend north India during April-May unless you are planning to go to Himalayas.
It'll be very hot there. Beaches of Goa & Kerala will be fine.
In any case you plan to come to Mumbai, don't hesitate in contacting me... we can meet over a cup of coffee.
Being a solo female traveller myself, I advise you to be more alert during your travel.
As mentioned by someone else, you can book your train tickets 90 days in advance. You need to chart out your plan.
India has diverse culture and has a lot to offer for sightseeing... be it beaches, hills or plains... You would surely enjoy your visit to India. However, I shall recommend a visit either during Feb - April or in Oct-Nov. Post April, Summers set in (at its Peak around 40-45 Degree Centigrade) and You may not enjoy your stay... during monsoons as well, enjoying beaches is not such a great idea. Though during Feb- April, or Oct - Nov, we have pleasant weather conditions suitable for travel to any part of the country.
Goa and Kerala are good options for the beaches and so is Andaman Islands. You might like to visit Gujrat (Gir Forest Reserve or Panna tiger Reserve) if you are looking at adventure in the forests... Rishikesh is a good place for spiritual tourism.. Dharamshala offers you the Buddhist experience and is Home to Dalai Lama... Leh - Ladakh offers you the best of Himalayas...
Just be cautious and careful while travelling solo in India.
Happy Travelling !