Romance seems to waft through the air on Capri, accompanied by the scent of lemon blossoms and fresh sea air. Leave the car on the mainland and join the masses who use water transport to flock to this dramatic Italian isle in the Bay of Naples. The luminescence of the Blue Grotto, the striking Faraglioni rocks and Palazzo al Mare add to the magical spell. Snorkel from Marina Piccola beach on the island's south side, believed to be the spot where Odysseus nearly succumbed to the Sirens' song.

Once a vital part of a mighty sea power, Positano is today a sophisticated resort on the central Amalfi Coast. Moorish-style architecture rises up steep slopes that gaze out on the Sirenuse Islands. Smart boutiques, selling fashions for visitors to display on Grand Beach, abound in the village. And it’s a great base for exploring the area—you can easily travel by boat to Capri, Ischia and the Grotta dello Smeraldo cave.

Most visitors to Fiumicino arrive by plane and immediately continue on to Rome, but—in addition to being home to Italy’s busiest airport—the coastal town doubles as a beach resort during the summer months. There’s also Ostia Antica archaeological site, which is just a short drive away.

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.

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.

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.

The misty hills, lush teakwood and sandalwood forests, and acres of tea and coffee plantations of Kodagu are, in a word, stunning. Also known as Coorg and dubbed "The Scotland of India," Kodagu is a postcard-perfect region of scattered villages and hamlets, which are the epitome of old-world charm. Kodagu is ideal for outdoor activities such as trekking, angling and white-water rafting, and major festivals like Keil Poldu (worship of weapons), Cauvery Shankaramana (return of the river goddess) and the Huttari (harvest) festival are a huge draw.

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.

Though often used as a gateway to nearby beach resorts, Thiruvananthapuram offers plenty of its own to reward travellers who stay. From South Indian cuisine and a sophisticated art scene to architectural treasures and historic landmarks, there’s so much to experience in Kerala’s capital.