Amritsar golden temple I guess, you won't need money for food as they serve food there for free all day and night. You can stay there as well for a few days not I don't think they would let you stay more then 3-4 days for free.
I suggest you try looking into couch surfing. Basically, you pay someone a small amount to sleep on their couch. I really like to do couch surfing because you get to meet new and interesting people, who are always happy to share amazing stories and even make you delicious meals if you are friendly.
With that said, where should I be looking for lodging arrangements for a 1-3 month stay in India.
I'm not down with hanging out in the slums or with the bedbugs either, a girl has her standards you know.