I see a lot of people having sink plungers next to their toilets. That might work for some things, but they might as well be using any other object.
When a pipe is clogged it's basically impossible to push anything through. It's stuck for a reason. The pushing motion might compress the clog somewhat, but it's really the pulling which causes it to move backwards and hopefully unclog it. This usually requires a plunger that actually seals the opening, or you'll have to push and pull until the water itself gets in motion to pull back the clog.
Another trick for unclogging toilets is to fill a bucket of water and pour it in a steady stream from as high as you can reach. Obviously start pouring from the seat and then lift the bucket while pouring. The gravity from the height will create a very strong narrow stream with enough force to pierce through or wash back the clog.