I used to travel by the local train every day. From Andheri to Churchgate. Trains are great equalizers in this city. No matter what your status in society, there is always some place for you in the most crowded compartments. Mumbai locals doors don't close. So people just hang out. It also provides an interesting way to commute during the monsoon seasons.
Another interesting part of the local trains is that there are separate compartments reserved for women. Mumbai is probably the safest city in India for women. However, it is still not safe for women to travel in a crowded local train with men. Groping is to be expected so watch out if you are in a crowded compartment. Try using the local trains. Buy 1st class tickets and board where the train line starts. It is the fastest way to travel in Mumbai. The tickets are affordable and the frequency of the trains is pretty high.