Shift to the "D" position, and try to drive gently, with smooth and gradual acceleration. Until the vehicle reaches a speed of 50-60 km/h or 30-37 mph, you should feel the gears shift at least twice (from first to second, and from second to third). These shifts should be made very smoothly, without jerks or slipping.
By "smooth shift" I mean it should be smooth but you should feel it at the moment the rpm drops down; it feels like the car slows down for a very short instance and then accelerates again. You also can note the transmission shifting moment by the slight change in the engine tone.
But when the transmission is extremely worn it may shift with quite a strong jerk, shudder or a delay (especially from first to second speed).
Driving at a speed of 40-50 km/h or 25-30 mph, if you press down the accelerator pedal for a few seconds, you should feel downshifting to the lower gear, if the automatic transmission works properly.