I would not recommend to anyone trying to live here on a tourist visa. You will not be able to rent an apartment in most states since that needs to be cleared though the Police Commissioner of the city you live in. You could probably share an apartment with someone else who has already filed the paperwork for the lease. You won't be able to register a SIM card or internet connection as a tourist. Well, not true. You can get a SIM card, but once your visa is up, it will expire. You will need to reapply to get your service turned back on. For me, I have sometimes had to wait six weeks for them to turn them back on. You won't be able to get a driver's license, buy a car or buy any property on a tourist visa. Even on an employment visa. I have been here nearly ten years and still have to register every year at the Foreigners Regional Registration Office (FRRO). Most expats call it the Foreigner's Rectal Reaming Office, or the fifth circle of hell. You are always missing a new piece of paper or they now want all your bank statements, or something else. Everything is up to whomever you are speaking to on that day. The next day you go back, you offer your papers and now that missing paper was not required but the date is wrong on another document. <sigh>
Another poster was worried about her dog. You CAN bring a dog. I brought both a dog and a cat with me from the US on an Employment visa. You need the USDA health certification or whatever agricultural paperwork your country provides for international domestic animals. You will need to find a vet who has experience doing this as everything has to be timed carefully. The shots given to the dog need to be a couple of weeks earlier, but not more than four weeks (or something like that - this was nearly ten years ago). I know people who have visited on tourist visas with dogs, but they typically have an "emotional therapy dog" certification. I don't know anyone who has stayed on a tourist visa and repeatedly come in and out with their dog. All you need is one customs guard who wants to make your life miserable and says you're missing some piece of paper no one has ever told you about. Trust me. This happens all the time.
Once you're here in India, most landlords seem fine with animals compared to landlords in the US. We had three dogs at one point (we've rescued two from the streets here) and never had a problem finding suitable housing (although neighbors can have a problem).
I will also say that there have been some limitations put on tourist visas a few years ago. Once you leave, you need to be out of the country for a period of time. I'm not sure how long, but I believe it was either 30 or 90 days. That could get expensive moving in and out every 90 days for a month.
Good luck!