I've been visiting, working, and loving India for many years. The country, the people, the beauty, and the majesty never cease to amaze me.
As far as all of the food precautions are concerned, they're valid but you should not let them prevent you from enjoying the cuisine. The problem has to do with microbes in water that are alien to your system. Therefore, raw veggies and stuff washed in water can be a problem. In all my trips I've only ever had one illness, and it was taken care of immediately with Indian medications that are calibrated for those exact bugs.
By mid November the weather will be on its way to good, depending on where you go.
Of the big cities Mumbai can be very expensive. Delhi and Kolkata are both more approachable. Delhi is full of fascinating things to see and do, but it remains a big city.
If you go to Agra the make sure to see Fathpur Sikri while you are there. It is an absolutely amazing site.
I am very partial to Rajasthan and have spent lots of time in Jodhpur. It is a wonderful city to visit, and not at all destroyed by development, etc. From there you can take trips up to Nagaur province to see some of the greatest monuments on the silk road. The Rajput people are wonderful (and Rajput men are utterly majestic).
You can also go down through the Aravali hills towards Udaipur, which is a very beautiful trip. If you like architecture the Jain temples at Ranakpur are breathtaking.
Someone mentioned Goa. I happen to love old Goa and have a friend who owns a hotel in Calangute. Southern Goa has been turned into a tourist trap but the Northern areas still have their hippy vibe. There are some amazing places to eat in Goa including the restaurant "La Plage".
If your worried about class issues, as some people here seem to be, then I would suggest doing something about class disparity at home. Don't go to India expecting over a billion people to conform to a Midwestern middle-class worldview.
It is said of Mother India that people who come to her will either love or hate the country. I have found this to be true. If you go with an open mind then you will have a great time I am sure.
Oh, do remember that you need a visa. This is a small hassle, but you can get it through your travel agent. Make sure that you get that settled well in advance. Don't leave it to the last minute or you won't be able to travel.