How can an object with zero potential and kinetic energy ever move?

Where we define the potential energy to be $0$ in classical mechanics is arbitrary. All that matters is the change in potential energy. Since you are just learning this stuff I will assume you are in an algebra based physics class, so I will avoid using calculus here.

Potential energies are nice because they tell us how much work is done by a conservative force. More specifically, the work done by a conservative force is given by $$W_{cons}=-\Delta U$$ where $U$ is the potential energy associated with that conservative force. This is useful because we also know that the net work done on an object determines its change in kinetic energy $$W_{net}=\Delta K$$

So, if we consider your case where we just have one conservative force acting on the object, we can conclude that $$\Delta K=-\Delta U$$

And so we see here that the only thing that determines how the motion of our object changes is just the change in potential energy. If we define the zero-point to be at the top of the cliff, then as the object falls its kinetic energy will grow and its potential energy will decrease and be negative when it hits the bottom of the cliff. If we define the zero-point to be at the bottom of the cliff then as the object falls its kinetic energy will grow and its potential energy will decrease to $0$ when it hits the bottom of the cliff. In either case the same thing happens because we have the same change in potential energy.

Also another question that was already discussed in SE (but i didn't find my answer there) is why do we talk about the potential energy of a system (ball+Earth) but kinetic energy of an object (ball)?

Typically in introductory physics classes we just consider the ball being in a uniform gravitational field and we don't even consider the Earth. However if you want to include the Earth in your system then when the ball falls the Earth will actually move slightly upwards to meet the ball. It will still be the case though that the work done by gravity on each object is related to the change in the potential energy of each object, which results in a change of kinetic energy.

The energy of an object consists of its internal energy (kinetic and potential energy possessed internally to an object at the atomic or molecular level) and external kinetic and potential energy. Its external kinetic and potential energy is always with respect to some external frame of reference.

The ball on the surface of the earth may have zero potential energy with respect to the surface of the earth, but has gravitational potential energy with respect to the center of the earth. It doesn’t move because the force of gravity downward on the ball is equal and opposite to the force exerted upward on it by the ground. But imagine if there was a trench that went from one side of the earth to the other. If the ball were positioned over the hole and allowed to fall, it would accelerate toward the center of the earth losing potential energy and gaining kinetic energy, the latter being a maximum at the center where its potential energy is zero. Then it would decelerate as it goes past the center as it gains potential energy and loses kinetic energy, ultimately stopping at the other side (ignoring air resistance).

Even the kinetic energy of the object is relative to the frame of reference where the velocity is measured. If you are in a car moving at velocity $v$ with respect to the road, it has zero kinetic energy in your frame of reference. It is not “moving” in your frame of reference, but is moving in the frame of reference of a person standing on the road. To that person the car has kinetic energy. The person standing on the road is “moving” with respect to your frame of reference in the car, so that person has kinetic energy with respect to you in the car. The person standing on the road has kinetic energy with respect to a frame of reference other than the earth (since the earth is rotating). And so on…

Bottom line, everything can be considered to have kinetic energy (be moving) with respect to some frame of reference and have potential energy (due to its position) with respect to some frame of reference.

Hope this help.

I am not sure how to ask this question but I am learning about potential energy (high school physics) and from the deffinition of a potential energy (energy stored in an object with the potential to convert into other type of energy)

this definition of potential energy, sounds more like the energy itself. More below

i don't understand how for eg. when we have an object (let's say a ball) on the ground, it has zero kinetic energy and also zero potential energy and now let's say the ball starts falling of a cliff so it will be gaining kinetic energy but when it had no potential energy, how can it be now gaining energy?

indeed, in the planet+object system, the true 0 potential, would be better put at the center of gravity the earth if the reference frame is attached to the earth, if the reference frame is attached to the object, then it should be on the center of gravity of the object, and if the reference frame is neither, the 0 potential should be at the center of gravity of the object+earth system.

Also another question that was already discussed in SE (but i didn't find my answer there) is why do we talk about the potential energy of a system (ball+Earth) but kinetic energy of an object (ball)?

Because you need to define what type of potential energy we are talking about, in this case it's gravity (could be electrical for example)

We now see that potential energy is relative to a type of force (gravity), reference frame, and the final nail in the coffin of your definition of potential energy, is that it's not "stored in the object" . Potential energies depend on the spacial position of the object in the force field.

I hope all these elements will help you understand potential energy. Meanwhile your teacher probably gave you a simplified version of this that is enough to solve the problems you are supposed to solve for now. The main idea the teacher probably wants to convey is that energy is conserved, and that like voltage, we are more interested in differences in potential than absolute potentials.