In the south Kerala could be a little cooler due to the breeze. If the OP is in the south and needs some relief, a trip to Kodaikanal or Munnar might help and will be quite cool. I omitted Ooty since I think its too crowded nowadays. Citywise Coimbatore and Bangalore will be nicer too, though not much to see or do for the tourist.

Sadly $600 is not much in India nowadays; one would have gotten much further with it at the turn of the milennium. My reccomendation, keep the itin modest. See Delhi's sites and Agra, the Taj Mahal. But for a real contrast and to appreciate how diverse India is, go down south to Tamil Nadu or something like that (I may or may not be biased since I'm of Tamil origin). It'll be a different language with different customs, colors, sights, sounds and smells to boot, but you get the idea.

In other words, don't be too ambitious to try and see everything, but at the same time, don't limit yourself to one area. Most importantly, have a great time (and don't drink the water unless its bottled).