Mumbai (formerly known as Bombay), India, is famous for its chaotic streets. For bargains and people-watching, outdoor bazaars top the list of attractions. Popular waterfront destinations are Marine Drive, where visitors go to watch the sun set over the Arabian Sea, and the carnival-like Juhu Beach. More sightseeing options are the Gandhi Museum, in the leader's former home, and the cave temples of Elephanta Island. For tranquillity, Mumbai has many religious sites, lakes and parks.

Love garlic? You’ll love the food in Pune. Unlike other cuisines of the subcontinent, Pune cuisine relies heavily on the aromatic bulb. Soothe your palate with sweets like bhakarwadi, a pastry rolled with coriander, tamarind and sesame seeds, or cool off with a thick milkshake made with dried fruit.

On the banks of the Volga River, Kazan sits austerely, its cityscape a visual confection of architecture that somehow manages to be both playful and severe. No matter your spiritual inclination, you simply must visit the Temple of All Religions, a Technicolor cultural center built by artist Ildar Khanov. Though still a work in progress, the "temple" is a feast for the eyes—and the spirit.

Laid out by British architect Edwin Lutyens, the Indian capital is a striking modern metropolis. A gracious contrast to Old Delhi's winding streets, the grand avenues and stately buildings of New Delhi are rich with history and culture, from Gandhi's Delhi home (and the site of his assassination) to the tomb of Humayun, a complex of Mughal buildings reminiscent of the Taj Mahal. Chaotic traffic is best left to the locals. Negotiate a good price for taxis or travel on the new Delhi Metro.

Want a taste of being royal? Eat in Hyderabad, where culinary traditions have been passed down from the Nizam monarchy. Arabic, Turkish and Mughlai influences are easily recognisable. The city is famous for its rich, aromatic biryani made with lamb, chicken or vegetables and served with fragrant basmati rice. Satisfy your sweet tooth with double-ka-meetha, a bread pudding.

Known as both the "Garden City" and "The Silicon Valley of India," Bangalore (officially "Bengaluru") is a techie’s paradise, boasting the highest concentration of IT companies in the country. When you’re done geeking out, there are plenty of gardens, museums, natural features, palaces and temples to fill your dance card. Visit Vidhana Soudha, Cubbon Park and the Ulsoor Lake of Bangalore, well known for its beautiful locales and boating facilities. Bangalore is also a major centre of Indian classical music and dance, and of vivid, cutting-edge nightlife.

The people of Tamil Nadu consider providing food to others a service to humanity. Thus the service in the state capital, Chennai, is first-rate. Treat your senses to some of the richest South Indian flavours in traditional dishes like sambar, rasam, fish curry or kootu. And don't forget to have a cup of full-bodied Tamil coffee, enhanced with chicory—no visit is complete without it.

This former British colonial stronghold boasts evidence of over two millennia of habitation, with ornate, architecturally diverse buildings, ranging from crumbing ruins to Victorian treasures. Home to lively festivals and a vibrant artistic community, clamorous markets and packed temples, this city is crowded and polluted, but ultimately invigorating.