Here are the points you brought up as unclear, in the order you mention them

a) "when a mass moves it acquires kinetic energy"
Yes you can calculate it exactly, the kinetic energy is 1/2 mv**2.
However the velocity v is always relative to some arbitrary coordinate system, it is a geometric concept, not a physical one. The kinetic energy of a falling rock is only apparent when it strikes the ground. To an insect standing on the rock, unaware of the ground, there is no way to determine that kinetic energy until after the collision.

b)The acceleration has nothing to do with the kinetic energy. The kinetic energy is determined by the speed of the moving mass, by 1/2mv**2. It doesn't matter if the acceleration leading to that velocity was high or low, quick or spread out over time. A 1 kilogram mass moving at 10 meters per second has an kinetic energy of 50 Joules. It doesn't matter if that mass fell on earth where the acceleration is 9.8 meters per second squared, or on the moon where it is 1/6 as much. The velocity matters, not the acceleration. But remember, velocity = acceleration x time. On the moon it would have to fall for six times as long a time to reach the same speed as on earth.

As for your idea of a "force field", there is no such thing. A force is what happens when a mass is accelerated, either by gravity, or a rocket motor, or because someone shoves on it. No field is needed. A force is something the mass does, not something that happens to the mass. A force is a mass times an acceleration, so the force acting on a 1 kg stone falling in the earth's gravity is F = ma = 9.8 Newtons. A 1000 kg automobile falling alongside it experiences a force of 9800 Newtons. If these objects were falling on the moon the forces on both would be one sixth as much. You will note the force is different for the stone and the car because the masses are different, not because "the field" is different. They are falling together, at the exact same speed, accelerating in exactly the same way. Only the mass is different.

By the way, a "Newton" is defined as 1 kg meter per second per second, or kg-m/s**2. A force acting on a mass over a distance is what gives that mass its energy. So a force of one Newton acting on a mass of 1 kg over a distance of 1 meter gives you an energy of one Joule.

Our one kilogram stone falling off a 100 meter building will have experienced a force of 9.8 Newtons across a distance of 100 meters, so it will have a kinetic energy when it hits of 980 Joules. We know the energy of a moving object is 1/2mv**2, so solving our 1 kg mass for v we get v = SQRT (2x 980) = 44.3 m/s. We know that the definition of velocity is acceleration times time, so 44.3 = 9.8t, so t = 4.5 s. Calculus can be used on these equations to reveal that the distance traveled d = 1/2 at**2. So going to our example of a 1 Kg brick falling off a 100 m building, the time it takes to reach the ground is 4.5 seconds, the speed when it hits is 44.3 m/s, the energy it releases on impact is 980 Joules. The equations can all be used to check each other, and we can physically measure the mass with a scale, the distances with a ruler, the time with a clock. The energy ties it all together.
Energy, matter, space and time. That's all you need, no spooky forces or magic energy reservoirs, or Jeddi force fields..

And one more thing, there is no such thing as a "field". The field is a mathematical concept obeying certain rules that allows us to calculate what those forces will be. Mathematically speaking, it is a 'three dimensional vector space' not an object or thing. They are useful in calculating gravity and electromagnetic problems, just like latitude and longitude are useful for navigation. But it has no real existence itself. When a physicist says something like a gravitational or magnetic field he is really referring to a mathematical equation that yields a force vector ponting towards or away from the center and with a magnitude equal to a constant times 1/distance from the center squared when a test charge or mass is placed at any point within it. It is a mathematical model, not a thing like a magic cloud with special properties. It's all in your mind.

Four centuries ago Newton thought of it as a thing that somehow powered other things to move, he called it "action at a distance" and he was (correctly, as it turns out) very suspicious of that explanation. Einstein later showed us it was a geometrical phenomenon, caused by the curvature of space-time. The field does not "give" energy to objects moving through it, or it would eventually have to run out of energy, right?

To summarize, in Newtonian mechanics, the fundamental quantities are matter (m), energy (E), space (d), and time (t). The units of each are kilograms, Joules, meters, and seconds.

velocity: v =d/t
acceleration: a = v/t
distance d = 1/2at**2
force: F = ma
Energy E = Fd = 1/2mv**2

and one we haven't discussed, but which can come in very handy, momentum p = mv