The city is best visited not for its monuments (it doesn't have too many) but for its ambience and vibe. The best way to start experiencing Mumbai would be to hop onto a ferry in the morning at the Gateway of India and head towards the Elephanta Caves. 9 km from Mumbai, the journey takes roughly an hour and on the way you can admire Mumbai's skyscrapers and the grand Taj Mahal Hotel.
These temple caves are dedicated to Lord Shiva and most of the people peg their origin somewhere between 450 and 750 AD. The caves are populated by sculptures of Hindu gods, situated in a large hall supported by pillars. Especially impressive is the statue of the three-headed Shiva, the Trimurti - as Creator, Destroyer and Protector. Regular dance and music performances are held here.
When you arrive back at the shore, hang around at the Gateway of India, buzzing with evening strollers and ice cream vendors. The gateway was built in 1911, and was the final point of departure for the British in 1948, from where they sailed back to England.
From here, stroll down the sharp arc of Marine Drive, offering uninterrupted views of the sea on one side and Mumbai's prominent skyline on the other. Night would fall by the time the drive ends at the colourful Chowpatty Beach on the south. The sea here is too dirty for a dip, but come here for the atmosphere. Eat a bhelpuri washed down with ice cream and watch children riding the Ferris wheel, people getting massages or just sitting around. Crowded, chaotic, and what fun.
Explore the districts of Old Mumbai, populated with lovely Gothic-Victorian edifices. Walk through the Colaba / Fort area to see the Victoria Terminus (recently renamed as the Chhatrapati Shivaji Terminus but known primarily as VT). It is one of the most extravagant works of the British, modelled after St. Pancras Station in London. A cornucopia of numerous architectural styles, it has domes and spires, minarets and turrets, columns and sculptures of a British lion and an Indian tiger at the entrance.
Situated in the heart of the Fort area is the Flora Fountain (renamed as Hutatma Chowk) with a sculpture of the Roman goddess Flora. On your way to the fountain, you will see more monuments dating from the British era such as the High Court building, the neoclassical Army & Navy Building and the Horniman Circle, a splendid colonnaded building with spiral staircases and marble statues.
Facing the circle is the St. Thomas' Cathedral, the oldest Raj-era building in Mumbai, dating 1712. Also visit the Asiatic Society Library here, built in the eighteenth century, housing teak bookshelves and wrought iron balconies. The library houses 800,000 books and about 10,000 rare manuscripts, including the first edition of Dante's 'Divine Comedy' that the society famously refused to sell to Mussolini.
What you will also see in this area are lovely synagogues including the pink and blue Kenneth Eliyahoo Synagogue, and Parsi Fire Temples, built in the late eighteenth century, their fronts rimmed with huge griffins.
Fort is also Mumbai's cultural district with the Prince of Wales Museum (renamed as Chhatrapati Shivaji Vaastu Sangrahalaya), set in an Indo-Sarcenic building and housing a collection that charts Indian history and culture, starting from the 3500 BC Indus Valley civilization. The Jehangir Art Gallery here has works by some of India's eminent contemporary artists. Also visit the National Gallery of Modern Art.