The traditional Thai greeting is not a handshake but a wai, where the hands are brought together in front of the chin. The younger or lower-ranking person wais first, but don't let this stand in your way if you feel someone merits a wai.
The Thai greeting referred to as the wai (Thai: ไหว้, pronounced [wâːj]) consists of a slight bow, with the palms pressed together in a prayer-like fashion.