As mentioned by others it is a part of the game. To understand it better you need to realize that football is a contact sport (not as much as rugby but still) and it's physically demanding. For that very reason players (especially in amateur divisions) will use physical intimidation as a way of winning the ball. Keep in mind that, going into a tackle if you are thinking that you might be hurt or whatever, you've pretty much already lost the ball. :)
That being said you do not need to start kicking people around, or strangle them by hanging on their shirts. In terms of pushing I believe you mean shoulder challenge whilst running. In that scenario it's very important to take stable steps, so that you have a solid balance when you take on the player. See to that the contact happens in your terms, as in be mentally and mechanically ready for the collision, and try to make sure that you go in with your hip or your shoulder. These are stronger joints, that way hopefully the momentum of the collision will not get your off your balance.
If you are referring to pushing while keeping/shielding the ball, then you might need to get a bit dirty. A lot of players will play rough in an attempt to intimidate opponents from reaching to the ball. Try to tilt his/her balance by pushing your opponent off balance. It helps trying to pay attention to how stable his/her stance is and how his/her weight is distributed. Trivial example; if the opponent has one leg on the ball and using his body to shield the ball, you obviously want him to move his pivot leg to block your movement. WHile that leg is in motion, the player is off balance and will most likely lose the ball if you push in a bit yourself.
These are my two cents, hope I didn't misunderstand your question. Good luck and keep in mind to have fun out there in the field. That's what matters really ;)