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.
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".
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...
Varanasi (aka. Benares) is one of the oldest living cities in the world! The city is very much alive and the center of the Hindu religion. People come here to pray, to wash in the river Ganges and to get cremated. For tourists this city is a highlight of any visit to India, but please show respect.
Mumbai, formerly known as Bombay, is the economic centre of India and a world-city. There are lots of places to visit including the Gateway To India and Bollywood. If you have some money to spend, stay at the Taj Mahal hotel... a place with tradition...