New Delhi is the capital of India and one of the largest cities in the world. The old city is quite picturesque - as is most of India. Unfortunately Delhi is also the most polluted city in the world and not a safe place for women alone at night...
Agra is world-famous for one of the most beautiful buildings in the world: the Taj Mahal. The completely symmetrical Mausoleum is made of white marble with lots and lots of precious stones built in. If in Agra, don´t forget to visit the red fort as well... Or dine on a roof...
Jaipur is capital of Rajasthan. It takes it's nickname, the Pink City, from pink colored buildings incl. Wind Palace. Nearby is the beautiful Amer/Amber fort/palace, that you could enter on the back of an elephant... Delhi, Agra and Jaipur together form the "Golden Triangle".
Kochi (Cochin) is the largest city in Kerala, one of the more modern, most educated states in India. Tourist will like it that most women still wear sarees and punjabis, not jeans. Kathakali dance is interesting, but very very very slow.
The backwaters are wetlands in Kerala, close to Aleppey. You can rent a comfortable boat for one or more days/nights and let the landscape pass by. The crew will prepare your diner, you take care of your pictures and the mosquitos.
Kolkata (Calcutta) is one of the three gigantic cities in India. It is probably the poorest, where riksjas can still be on foot. Mother Theresa lived three for many years to care for a needy population.
Mysore, also known as Mysuru, is a pleasant city in Southern India with markets and the Sultan´s palace. Mysore is a great stop when visiting Ooty or Goa as well.
Pushkar is a city in Rajasthan, a pilgrimage site for Hindus and Sikhs with a sacred lake and the largest Camel + Elephant market in the world.