"Please go up the stairs."
One simple question you can ask to determine whether に or を is used:
After you finish the movement action, are you staying in that place? If yes, then に, and if no, then を. (If it is not a movement action altogether, then most probably you need a で.)
乗る means "to board/ride on." After boarding on the train, you stay on the train. So it is に.
After crossing the bridge, you do not stay on the bridge. You stay at the opposite side. So we use を.
After you turn at the corner, you do not stay at the corner. you stay on the next road. So を.
After you climb the mountain, you can either stay in the mountain, or stay at the opposite side of the mountain. So depending on your destination, you can use に or を.
Play is not a movement verb, so use で.