Before dismissing the idea of hiring a car in Mumbai completely, it is worth bearing in mind Mumbai is really a city of cities, and that much of the more recent development has been concentrated around Navi Mumbai (New Mumbai). This area is quite distinct from the older parts of the city.
Mumbai is also famous for its notoriously congested trains, although you should at least expect to travel on the inside of the Mumbai Metro. Back on the surface, local transport is chaotic and slow-moving, but many visitors will prefer to leave the driving to the operator of whatever taxi or bus they are in.
There are a few day trips which are worth taking outside the city, and you might want to do this by private tour or by organised bus tour, but a self drive car hire isn’t really particularly practical or desirable here.