If electrons move inside the wire, surely there is an electric force that cause this movement. Does this force change in various points of the simple electric circuit as the electrons move from the negative terminal to the positive, or does it stays the same all of the time??
Yes, there is an electric field within the resistor material, and this causes a force on the free carriers in the material.
For a deeper explanation, you should investigate the Drude model of electrical conduction in metals.
Does this force change in various points of the simple electric circuit as the electrons move from the negative terminal to the positive, or does it stays the same all of the time??
If the material is uniform, the field will be uniform.
If you had, say one piece of copper wire ($\rho=1.68\times{}10^{-8}\ \mathrm{\Omega{}\cdot{}m}$) and one equal-sized piece of steel wire ($\rho\approx{}6.9\times{}10^{-7}\ \mathrm{\Omega{}\cdot{}m}$) connected in series between your battery terminals, then the field in the copper wire will be lower than in the steel wire.
If the force changes, why doesn't the current in various points change in series circuits??
If the current were different in different parts of the circuit, then charge would have to build up somewhere in the circuit. This accumulated charge would repel other carriers of the same polarity, produce an opposing field, until the current equalized. This would be a dynamic effect, and when we talk about the dc steady-state solution to a circuit, we mean the solution after all such dynamic effects have settled down.
In my example with two different types of conductors, the materials have different resistivity, so the field (and resulting force) must be different exactly to produce equal currents in the two materials.
why the greater the difference in electric potential the greater the current??
Potential difference is the integral of field
$$V_{ba} = \int^b_a \vec{E}\cdot\mathrm{d}\vec{l}$$
So in order to increase the potential between two points ($a$ and $b$) you must increase the field strength along the path between them. The stronger field produces a stronger force on the free carriers, and so you get a larger current.
