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Near Kathmandu there's a place called Boudha.
Khenchen Thrangu Rinpoche, one of the most famous lamas in Kagyu lineage, has his teaching center there. Check out http://www.rinpoche.com/boudha18.htm
You can go as a tourist first and then figure out permanent status.
The Kathmandu Valley is a major place of pilgrimage for Buddhists. Around the city of Kathmandu are ancient Buddhist stupas such as the Great Boudhanath Stupa. The area surrounding the stupa which is known as “Boudha” has over sixty Tibetan Buddhist monasteries. There are many teachers and high lamas, either living in Boudha or visiting. Due to this, many students of Buddhism from all over the world come to stay here to receive teachings. Some monasteries provide situations for study in Higher Buddhist Philosophy. Khenchen Thrangu Rinpoche’s main monastery known as Thrangu Tashi Choling is located in Boudha close to the Boudha Stupa.
In the hills nearby Boudha, Phullahari Monastery, the seat of HE Jamgon Kongrul Rinpoche is located.
In the area of Swayambunath, is the Swayambunath Stupa also known as “The Monkey Temple”. Located on top of a hill, the whole hill itself is considered to be a self arisen stupa. According to legend, Lord Manjushri cut into the lake that was once the Kathmandu Valley and as the water was set free the hill rose up simultaneously. Thrangu Tara Abbey, Thrangu Rinpoche’s nunnery for Buddhist nuns is in Swayambu. There are many monasteries in the Swayambu area including Benchen Monastery, the home of Tenga Rinpoche and Sangye Nyenpa Rinpoche.
Among the many places of pilgrimage is Namo Buddha, one of the most important, where, in a previous life as a prince, the Buddha gave his body to a starving tigress and her cubs. At Namo Buddha, Thrangu Rinpoche has a temple, school for young monks, a college for Higher Buddhist Studies and other projects.
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To start with, here are general information on visa. Beware that the specificity depends on your citizenship (basically, what country issued your passport). Just some general points are:
For these reason, I concur with Andrei: try on a tourist visa. And I suggest you also give a try to the Himachal Pradesh in India.
In Nepal, if you don't speak Tibetan, I believe you can study in English at the following places. I'm not entirely sure, though.
In Buddhanath, there are a few Shedra (i.e. "places of teachings") open to Westerners. The question is always: do they teach in English?
In India, there is the Institute of Buddhist Studies and Dialectics.
I am sorry I cannot help you much with this. But do not think that you have to live in one of these countries to be a monk in a Tibetan tradition. There are places in the West.