I think it's easier to understand the physics when considering the microscopic building blocks of matter. The problem with this approach is that for a complete understanding one needs quantum theory, but one can get along in classical electromagnetism to some extent with classical qualitative pictures and then adapting some macroscopic parameters from measurement.
In electrostatics you deal with electric charges at rest. Electric charge is a fundamental quantity of matter discovered through the observation that bodies interact through the electromagnetic interaction. If you have a small charged body with charge ##q_1## (small in the sense that you can treat it to good approximation as a "point particle") at rest at one place there is a force acting on some other such point charge ##q_2## at a different place, and this force's magnitude is found to be proportional to the product of the two charges and proportional to ##1/r^2##, where ##r## is the distance between the charges. The proportionality constant defines the unit of charges. In the international system of units charges are measured in the unit Coulomb, and the proportionality constant is named ##1/(4 \pi \epsilon_0)##, where ##\epsilon_0 \simeq 8.854 \cdot 10^{-12} \frac{\text{C}^2}{\text{N} \text{m}^2}##, i.e., the magnitude of the force between the charges becomes
$$|\vec{F}|=\frac{|q_1 q_2|}{4 \pi \epsilon_0 r^2}.$$
The direction of the force is always along the straight line connecting the locations of the charges and the force is repelling if the charges have equal sign and attracting if they have opposite signs.
Then there is the important concept of the electric field. The idea is that there is not simply "an action at a distance" but that the force on the charge ##q_2## is due to the electric field present due to the presence of the charge ##q_1##. That means you can write the magnitude of the force acting on ##q_2## as ##|\vec{F}|=|q_2| |\vec{E}|##, where ##\vec{E}## is the electric field at the position of ##q_2##. This makes the interaction between the charges a local effect, i.e., the charge ##q_1## has its electric field around it and the force on ##q_2## is due to the presence of this electric field, and the strength is determined by the strength of the field at the position of ##q_2##.
From the point of view of electricity and magnetism there are roughly two kinds of matter, conductors and dielectrics. All matter consists of positively charged atomic nuclei surrounded by negatively charged electrons. In a dielectric the electrons and the positive nuclei are quite strictly bound together. If you bring now some additional electric charge close to this matter, all that happens is that through the electrostatic force you shift the electrons a bit in one and the nuclei a bit in another direction, but due to the pretty strong binding that's it. The electrons cannot move much relative to the nuclei. If you have some positive and negative charges just displaced a bit in opposite directions the main effect is what's called a dipole. Thus the charge brought from outside to a dielectric leads to the formation of a distribution of little dipoles. One says the dieelectric gets polarized, which weakens the electric field inside the dieelectric compared to the field which would be there in vacuum. If the electric field due to the additional charges is not too large, you can assume that the polarization is proportional to the electric field.
In a conductor some of the electrons around the nuclei are only very weakly bound within the material, i.e., those electrons can move pretty freely within it. If you now bring an additional charge close to it, the electrons start to move due to the electrostatic force. This situation is now not anymore electrostatics, but since there is also some friction in the medium the electrons after some time come to rest. The final positions of the electrons is such that there is some redistribution of charges along the surface of the conductor, leading to an induced electric field such that the net force on each electron is 0. That's induction.
Usually matter is electrically more or less neutral, but if you now rub two different materials against each other it may be that one of the material attracts the electrons on the contact surface more than the other. Then some electrons go from one material to the other. This results in a net charge of the bodies. Since electric charge is always strictly conserved the one body, who gave off some of its electrons, now has a positive net charge ##Q>0## and the other body, which took the electrons carries precisely the opposite charge ##-Q<0##.
I hope this qualitative picture helps to understand better the phenomena you asked about: On a very fundamental level everything can be understood from the original observation of electric interactions, i.e., that there's a property of matter described as electric charge carried by the particles making up this matter, and that the presence of electric charge implies that there's an electric field around it, and due to this electric field an electric force acts on other electric charges.
