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...
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.
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...
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.
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".
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...
Bandhavgarh National Park is a national park and Tiger Reserve in India. This park has a large biodiversity. The density of the tiger population (8 tigers per km2) is one of the highest known in India. It also hosts leopards and deer.
Bodhgaya is the place where Siddhartha Gautama attained enlightenment while sitting under the Bodhi tree. Herewith he turned into Lord Buddha ("the awakened one" or ‘the enlightened one’), the teacher with 250 million followers.