Force Causes Acceleration
We know from Newton's First Law of motion -- the Law of Inertia -- that if the net force on an object is zero, the acceleration of that object will also be zero. An object at rest has zero acceleration. An object that continues to move in a straight line with constant velocity has zero acceleration.
If the net force on an object is not zero, we find that the object accelerates. More than that, we find that the acceration is proportional to the force. We can write this as
F ~ a
where the ~ means "is proportional to".
If we push on an object with a greater force, we find the object has a greater acceleration.