Amritsar is the capital of Punjab and center of the Sikh religion. The city offers several typical Indian buildings, but the Golden Temple is definitely the main attraction. Women may want to go shopping for Punjabi dresses. From here you can also take a funny tour to the Pakistani border...
Chennai (Madras) is a major city in the South.
Darjeeling is one of the most remote places in the Indian sub-continent, where it is located between Nepal and Bhutan. The area is known for mountains and tea.
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".
Jaisalmer is a desert city with high walls similar to a fortress. The picturesque city will leave a permanent impression on you. From here it is easy to organize a camel trip to the desert. Please be aware of cheaters! Rates charged by hotels for tours can be a multitude of what the actual rate is!!! Hotels and private accommodations do not own camels...
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.
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.
Leh is a small city in the Western Himalayas and one of the destinations if you like to visit Kashmir.