Fly into Bengaluru (also called Bangalore) and stay at one of the city’s several high-end hotels — you’ll enjoy researching the range of opulent options. Visit the National Gallery of Modern Art, housed in a stately mansion, and Tipu Sultan’s palace and fort, a unique example of Indo-Islamic architecture. Head to the Krishna Rajendra Market early in the morning for copper goods, spices and a walk through a bustling flower market.

Hire a luxury private train to take you around the region in style, beginning with a trip to nearby Mysuru (also called Mysore). Visit Mysore Palace, one of the city’s grandest. Shop at the Government Silk Weaving Factory for saris and other silk goods and try typical South Indian cuisine from idlis and masala dosas to bisi bele bath, a lentil and rice dish. Visit the nearby Kabini River jungle (Nagarahole National Park) on a Jeep safari to see leopards, elephants and Indian bison.

Next reboard your train for Hampi, a 14th-century Hindu city that is a World Heritage site, second only to Beijing in size during the medieval era. The site has 1,600 monuments in 16 square miles, so hire a guide and give yourself a couple of days to explore.

Travel to the Badami caves, a well-preserved example of rock-cut architecture featuring Hindu, Jain and Buddhist temples carved into sandstone caves from the sixth to eighth centuries.

Board the train and head to Goa, where you can visit tropical spice plantations and 17th-century Portuguese churches. Don’t miss out on relaxing days soaking up sun on some of India’s most popular beaches, where surfing and beach snacks go hand in hand.