For first time visitors to Mumbai, unless you are an especially intrepid traveller, plan well if  attempting to move around on one's own. If you are in a group, it is that much easier. Otherwise have a reliable local contact, or else depend on a reliable  hotel/travel operator.  Mumbai is an interesting city and there's lots one can do/experience that is different from many other parts of the world. But, apart from the weather factor (barring mid Nov to mid March) there are also the large crowds everywhere and the attendant problems to overcome. Plus, traffic is quite chaotic and takes rather some time to get used to.

The best bet,unless there' is some local friend/contact, is to hire a car with driver for the entire duration of the visit.  It is not very expensive in terms of Euro or U.S dollar. The flipside of this is of course that one somewhat misses out on an 'authentic' experience. There are broadly three forms of public transport in Mumbai: trains, taxicabs (plus threewheeler autorickshaws in the suburbs) and buses.

The trains are quite efficient and the system is quite easy to understand.However the major drawback is the  huge crowds especially at rush hours and, the fact that the doors do not close while the train is in motion. Also, they cover only certain parts of the city and train travel has to be often supplemented by other modes of travel including walking. There are a few different lines so refer to train map before you get on the train. Second class tickets cost maximum 10 Rs. if you're travelling only inside Mumbai. First class tickets are 8~10 times more expansive. First class section have cushion seats and is less crowded. First class section have yellow and red slant stripes painted outside near the door. There's also female only compartment.

The buses are not very convenient except to connect between a few specific places.There are few guides and  the bus indicator systems are not  user friendly for the outside visitor.

As for taxis except a few 'company operated' taxis like Meru Cabs, Easy Cabs etc. the normal 'black and yellow' taxicabs do not have airconditioning and are small, not particularly comfortable vehicles. Caveat; as far as possible, opt for the more modern taxicabs which run on electronic meters. The autorickshaws in the suburbs are in a class by themseves- cheaper than taxicabs, but  less comfortable. Taxis don't have navigation (not sure about company operated ones) and drivers don't know the place if it's not famous. If you have GPS and google map, you can guide them, but if  you don't, you can have difficult time reaching the destination unless you're familiar with the area or the driver ask the local pedestrian and luckily he knows the place. Depending on the starting point and destination, it might be better to use the Bandra-Worli Sea Link. You'll have to pay extra 55 Rs. at the toll gate. 

 

Overall advice; decide what you want to do in the city and accordingly decide on location of stay. If you are planning to stay at an up-scale hotel in Navi Mumbai ( the satellite twin city of Mumbai), you can look for options like Fortune amongst others.  Getting around the city would have to be dovetailed around this. And always remember to allow sufficient additional time to get through day time traffic if one has to adhere to specific time deadlines, such as for reaching the airport.