You can use a plunger both ways. That is, you can use it to force the goods down the hole, and you can also set up a suction with the business end to pull it back. That may be necessary from time to time to help loosen the clog.
And don’t push air with the plunger, unless you have one of those newer bulb-shaped ones that works with a much larger bolus of air than the old-fashioned cup-shaped plungers. Those do better when you lower them into the bowl so there is no air at all in the plunger, and you’re using water pressure alone to set up the positive (or negative) pressure (or suction).
Rinse off the rubber end with the hottest water you can draw in the bathtub after you’re done, and then rinse out the tub, too, again with hot water.
PS: Don’t jump the gun and attempt to flush again until the water level has been reduced to way below normal in the bowl.