That means that the object is moving - that it doesn't stay in the same place all the time.
when the object is at rest
Nothing will happen to the object's motion, it will continue along its path.
When a moving object is pushed in the direction of its motion the speed of the object increases
Gravity I guess :-)
freefall
when an object changes its motion it is because a new force that is greater than the original force has been applied to the object and that object changes its motion unless the force is in the same direction as the original force, which would then imply that the object stays in the same direction of motion, but moves at a greater speed in that direction.
The object would accelerate faster.
Nothing will happen. Nothing will happen. I think.
when can you say that the object is in motion
Its speed, or direction, or both, change. That's called "acceleration".
That is the object's 'speed'.
That is the object's 'speed'.