I was born in Mumbai and have been living here since then so it is home to me. Most travellers to Mumbai are not too impressed with the city as it is a very crowded place with a lot of air and noise pollution. The city is mainly used as a gateway to India and also has a monument built on the occasion of the arrival of King George V to attend the Delhi Durbar. Most tourists come to Mumbai as a starting point or exit point on their way to or from India and hardly get to see the city and what it has to offer. I would suggest at least a few days to explore and see the city for all it has to offer the traveller.
There are so many things to see in this amazing city full of history. Just to get you started, here's an itinerary that can be done in one day and covers a lot of the city sites and sights.
If you are exploring Mumbai on your own, visit the Gateway of India and from there you can take a boat over to Elephanta Island for a look at the famous caves . Tickets can be bought in front of the Gateway from one of the booths selling them. There is a free guide inside the caves complex from morning to afternoon (just inquire with the security guard for his/her whereabouts).
After you get back from the islands you can carry on walking towards the Regal Theatre and see the Chattrapati Shivaji Maharaj Vastusangrahalaya (erstwhile Prince of Wales Museum) where you can spend a couple of hours absorbing India's rich past as well as art galleries such as the National Gallery of Modern Art and the Jehangir Art Gallery for some interesting art exhibitions. Then carry on towards the other side of the university of Mumbai buildings and see the Rajabai Clock Tower , Library and Convocation Hall. Also see the High Court of Mumbai buildings. Then if you look across, there will be a game of cricket going on in one of the famous playing grounds (maidans) called the Oval Maidan. Across the street is the Eros theatre and opposite that is the Churchgate railway station and in front of that is the Western Railway Headquarters as well as the Tourism office.
From there you can walk down to the Marine Drive promenade and take a long walk towards Chowpatty beach for a snack of Bhelpuri, Pav Bhaji and also some Indian ice cream called Kulfi. There are some other nice places to see in between like the Flora Fountain and Hutatma Chowk (Martyr's Square) and the St. Thomas' Cathedral that is very beautiful with a lot of nice sculptures inside it.
Nearby is the Bombay Stock Exchange building, the David Elaiyahoo Synagogue and the David Sassoon Library as well as the Elphinstone College buildings opposite it. The Town Hall building with the Horniman Circle gardens is also worth seeing behind the cathedral. You can visit Mani Bhavan (the house in which Mahatma Gandhi lived in Mumbai) which is now a museum devoted to his work and life as the father of the nation. Nearby is the F. D. Alpaiwalla Museum which has a lot of Iranian artefacts housed in the Khareghat Colony Hall and if you climb the steps leading up to the Hanging Gardens from the outside then you will be able to get a good view of the Queen's Necklace from above. On the way down you can visit a very elaborately carved Jain temple, or even go to the famous Banganga Tank and then come down towards the Governor of Maharashtra's bungalow before reaching Chowpatty beach or go towards the Babulnath Temple. If you wish you can still explore some of the other parts of the city like the Dhobi Ghats at Mahalaxmi Railway Station (which is a public laundry where the washermen wash and bleach the clothes before drying and ironing them to give them to their customers at home), the Haji Ali Dargah in the middle of the sea reachable by a small concrete path in the sea, the Mahalaxmi Temple nearby is also worth visiting. Up north in the suburbs of Borivali there is the 2nd Century old Kanheri Caves which are about 100 in number and built on a hill inside the Sanjay Gandhi (Borivali) National Park which is the only National Park in the middle of a metropolitan city.
If you are interested in shopping then depending on what you want to shop for there are many specific markets like Zaveri Bazaar for gold and silver jewelry, Mangaldas Market for clothes, Chor Bazaar for antiques and Bandra Linking Road for lots of designer shops. Lots of other shopping arcades and malls have recently opened up in many areas so you can just walk into one for a quick look and buy whatever catches your fancy.
This is the erstwhile Prince of Wales Museum of Western India now known as the Chhatrapati Shivaji Maharaj Vastu Sangrahalaya in the Art District of Mumbai. It is a lovely museum with a lot of different artefacts from all over India and some from other countries too.
Jimmy Boy is a Parsi cuisine restaurant that serves the traditional wedding style food just the way it is served at a Parsi wedding feast. There are Parsi style vegetarian options also available just like they are at a Parsi wedding. It also has other a la carte dishes available separately. They also serve the other wise rarely available Raspberry, Ginger and Ice Cream Soda aerated drinks. Worth a visit if you want to savour Parsi style cooking.
This is the first Anglican Cathedral built by the British under the Church of North India. It has a lovely fountain built from funds by a Parsi philanthropist at the entrance. The stained glass windows depicting the three Saints Thomas, Gabriel and Michael are made in the Pre-Raphaelite style and are worth seeing in the St. Thomas Chapel to the right as one enters the cathedral. The stained glass windows in the main altar are also really beautiful. There are many fine white marble memorials to British soldiers and officers during the many wars who have been buried here or remembered through memorials here.