Hey Ritika, I would suggest Jaipur as it’s closer to Delhi & the weather in Jaipur would be just perfect in October. You could stay at hostels like Zostel or Moustache which is a great option if you are travelling solo and are on a budget.
I know being a so.o traveler, you might have some questions related to your safety. I would suggest you to please go to Harsil, uttarkashi. It is just 25km from Gangotri. You can also go for gaumukh and see the origin of Bhagirathi. It’s an overnight journey from Delhi.