If you are traveling in India you should definitively visit Bodhgaya, not so far away from Varanasi. This is one of the holiest cities in India where prince Siddharta Gautama sat in meditation under the buddhi tree until he reached enlightenment and became The Buddha. The Mahabodhi Temple was built around the same place where Buddha attained enlightenment. It is a pilgrimage place, mainly for Buddhist of course, and you'll be able to see the Buddhist temples of all the different buddhist traditions like the Tibetan Buddhism, Chinese Buddhism, Japanese Buddhism, Thai Buddhism and many others.
If you manage to visit Bodhgaya I would recommend to pass by The Root Institute and consider following one of their courses. I stayed there a few years ago to follow the 10 days introduction course. Unfortunately I can't remember much about the content of the course, I remember we have some analytical meditation sessions and discussions about Karma and Emptiness.
The facilities are quite good, the rooms are comfortable and the bathrooms are clean. I don't remember seeing the library like it is mentioned in their website but I do remember that they have a clinic at the entrance where they will give free consultations to the people from the region. I was actually very happy that they have this clinic cause I got sick, I was actually very sick for about 7 days out of the 10 days. It was very difficult to go to the classes and meditations with fever and pain in all my body but they did take good care of me. I was not the only one sick, there were 2 other guys sick, even sicker than I was, they spent the whole time in the clinic. I saw them only twice during the course.
The food is delicious vegetarion food and in the breakfast along with all the normal stuff like bread, mermelade, milk, eggs they will give you a delicious home made peanut butter.... hmmmm really good. I hope they haven't change that.
Its a small institute with only one gompa and space for about 50 people. But it seems that it has changed with the years, now they even have a huge praying wheel.
Hear I learned about the Kopan Monastery in Nepal, a beautiful monastery located near Katmandu where foreigners can go for a 1 month course in Buddhism. I also recommend you to go to Kopan but if you intend to go to Katmandu from Bodhgaya be very careful, it can be a very tough and challenging journey. Don't trust too much the tourist offices in Bodhgaya, specially those that want to sell you the direct bus ride to Katmandu were you'll need to take 3 buses and everything is "supposed" to be arranged in advance. Consider better the long journey with the train.