Question

According to Newton's third law, if you push on a wall, the wall ?

Answer 1

The wall will push back.

Answer 2

The wall will push back, in exactly the opposite direction, and with
exactly the same size force.

That's why the net force on the palm of your hand is zero, and that
in turn is the reason that your hand doesn't accelerate.

If you keep increasing the strength of your push, then eventually you
exceed the force that the wall is capable of delivering.  Then the wall
crumbles and falls, your hand accelerates in the direction you're pushing,
and the crowd goes wild !