There are tonnes of festivals celebrated in India. A few that I'd really love to celebrate in the country itself would be Diwali and Holi, since the whole atmosphere in India would be completely different. I think that next time I visit India, I'll try and plan the holiday so that it coincides with one of these festivals.
Which festivals would you love to celebrate in India, or which festivals would you travel to India for?
Diwali and Holi are excellent times to visit India. It helps that Diwali is in October/November and Holi is in March, which are both really pleasant months weather wise. However, you may want to be at the right place as well. Holi is celebrated in the northern part of India and not su much in the south. I grew up in south India, and the first time I celebrated Holi was in grad school in the United States

Ideally, you would want to be in Mathura or Vrindavan for the festival. These places are pretty close to Delhi. Diwali is a pan-Indian festival, but the sights you may see would vary depending on where you are in the country. In the north, people light many many lamps all over the city at night. In the south, this custom is not as common. But people celebrate with fireworks in the south. These are personal fireworks that every family buys and lights in front of their houses

So it is a lot of fun. Be aware that it can get rather noisy and there is always a fire hazard, so stay safe. If you have animals, they might not be very comfortable with the noise and smoke. Your best bet would be to be to get invited home by friends in India so that you can celebrate with them.
Personally, I love being in India during Navaratri. This is a nine-night festival held during the months of September-October, about two weeks before Diwali. It is a festival dedicated to the Mother Goddess. In Kolkata/Calcutta (Bengal) this period is called Durga Pujo. I have always wanted to visit Kolkata during Durga Puja. Every year, leading up to the festival, sculptors make huge statues of Kali, one facet of the Mother Goddess. The festivities last through the night. In Tamil Nadu, we celebrate with a doll display festival called Kolu. Throughout the year, we collect dolls made of stone, brick, porcelain, papier mache and other relatively solid materials. We display them for the nine-day Navaratri period by building elaborate sets of steps and arranging the dolls on them. If you are in south India during this season, you could visit the major temples to see the display of dolls. In Kerala, during this season, a phenomenal music concert called Navaratri Mandapam is conducted; it is a cultural revelation.