To ensure that the monks aren't heavily disturbed, there are various restrictions on visiting times. Things were slightly easier for me as I was not only making pilgrimage but also visiting a friend of mine who was a novice at Iveron (and has since become a priest-monk). Going to Mt. Athos, while an incredible experience, requires quite a bit of preparation: not only should you be familiar with Orthodox monastic and liturgical life, you probably will also either want to know Greek yourself or have someone with you who is fairly fluent in Greek. As such, a better starting point to exploring monasteries is to find one close to you where the monks actually speak your native language. :)
I should add that the book I previously mentioned, "The Mountain of Silence," provides a very approachable introduction to Orthodox monastic life (which is really the Orthodox Christian life lived in a particular mode), without requiring any religious background. The author himself is writing from the perspective of an anthropologist, although I think at the time he was also a lapsed Orthodox Christian. Suffice to say that a relatively non-religious friend of mine really enjoyed the book. :)
All that being said, more info on visiting Mt. Athos is here: http://www.athosfriends.org/PilgrimsGuide/planning/ I should note that Athos does not allow women visitors, but there are similar monastic communities for women. This is not at all intended to be sexist, but is part of the asceticism of Athonite monasticism in particular. Previously women have settled on Athos, and the monks simply let them live there -- they ended up leaving of their own accord. Also, the monks sheltered families of Jews during WW2, which of course included women.
