Locally - car, two wheeler, autorickshaw.
Public transport possible - buses crowded and not to clean, but cheap
Also, some cities, like Hyderabad and Madras have a suburban train system. Like the buses, crowded and not too clean. First class may be available. Useful if you are on the route.
Bicycle is hot and not too suited for crowded city traffic
Longer trips - Car - own or rental
Train - easy and fairly safe
Long range bus - convenient but not too safe or comfortable
Plane - pricey and not as reliable as one might expect. Changing schedules and delayed departures.