I've been round south east India during August. We chose this area as it is not affected by the monsoon at this time (I think it comes later) and it was actually a very pleasant time to travel. We went to Mysore, Bangalore, Hyderabad, Madurai and a couple of hill station (Ooty and Kodai) when the weather was posivitely English.
While I think these places are definitely doable during August, I would hesitate to take a very small child there or indeed any where else in India. My main worry would be road safety, every time we've been in India we've seen the aftermath of some pretty horrendous road accidents, we've also been on some very, very scarey bus rides. Perhaps things have changed (I haven't been for eight years) however, when we were last there, none of the cars had back seat safety belts, so putting your dd in a car seat would be out of the question.
You can of course travel to a lot of places by train, but if you want to travel to some of the hill stations you will definitely have to get a bus or hire a car with a driver and I don't think I'd do this with a 2 yr old.
I'm still planning to take my kids there but not until they're at least 5 or 6 and then that will be for visiting friends in Delhi or perhaps staying in a hotel in Goa/Kerala.
Good luck!