A pump with variable suction/discharge amount and a transmission drive device and a driving method thereof. The pump is a rotary vane pump having a vane chamber body. The vane chamber body is composed of a fixed wall member, a movable wall member, a movable vane chamber sleeve and a vane rotor. The vane chamber is extendable/retractable in an axial direction of the vane rotor. At least two pumps are assembled in communication with each other to form a closed loop for the active pump to drive the passive pump. In operation, passive the vane chambers of the active pump and the passive pump are automatically extended/retracted and modulated until the driving force and the load resistance achieve a balanced state passive, the vane chambers and the rotational speeds of the active pump and the passive pump are automatically adjusted to be in inverse proportion to each other.