A flanged plunger seals the pan outlet properly; a flat cup plunger does not and mostly just splashes. Push and pull steadily rather than hammering — you want to move the water column back and forth, not force it one way.
A closet auger is the right tool if plunging fails. It feeds a coiled cable around the trap with a plastic sleeve protecting the porcelain. Never push a coat hanger or screwdriver into the pan — you will scratch the glaze and create the rough surface the next blockage forms on.
If the toilet clears but drains slowly, or if other fixtures gurgle when you flush, the problem is downstream in the soil pipe or underground drain and needs rodding or jetting.