Flashes are triggered from within the camera when you trip the shutter release button. If the flash has its "own trigger", chances are that trigger is the "TEST" button. Not all flashes have that feature. Flashes are supposed to fire at the same time the shutter opens. Triggering the flash manually isn't the "normal" way to do it.
If you connect the flash to the camera (through the hot shoe), when you take a shot, the flash should automatically fire if the settings (and conditions) are right. To force the flash to fire, set camera to manual mode and flash to auto mode. Set camera to f/16, 1/60s. Stay at least 10 feet away from your subject. Take the shot. If the picture is too bright, step back and try again. If too dark, go nearer and try again.