Halacha requires washing one's hands after taking a haircut and after cutting his nails. The procedure for this washing is the same procedure required when one washes his hands upon waking in the morning: he washes both hands three times in alternating fashion. Unlike the morning washing, however, the washing after a haircut or nail-cutting does not require the recitation of a Beracha.
A person is not required to leave the barbershop or room where he cut his nails before washing his hands. Upon leaving a bathhouse or Mikveh, Halacha indeed requires one to leave the area before washing his hands, due to the presence of unclothed people, sweat and odor in these areas. Similarly, after visiting a cemetery one washes his hands specifically after he leaves the cemetery, as he may not wash while still exposed to the "impurity" of the graveyard. In the case of a haircut or nail-cutting, however, we do not deal with any filth or impurity, and as such Halacha permits hand-washing in the same room as one had his hair or nails cut. In fact, it is preferable to wash one's hands in the same room, in order not to delay the washing any longer than necessary. This is the ruling of Rav Efrayim Greenblat (contemporary scholar in Memphis), in his work Rivevot Efrayim (vol. 1, 7:3).
Summary: After one has his hair or nails cut, he must wash his hands three times in alternating fashion, without a Beracha. He should wash as soon after the cutting as possible, even in the same room in which his hair or nails were cut.
