How do electrons get a charge?

Your question touches the question of ontology in particle physics. Historically we are used to be thinking of particles as tiny independent entities that behave according to some laws of motion. This stems from the atomistic theory of matter, which was developed some two thousand years ago from the starting point of what would happen if we could split matter in ever smaller parts. The old Greeks came to the conclusion that there had to be a limit to that splitting, hence the atom hypothesis was born.

This was just a philosophical idea, of course, until around the beginning of the 19th century we learned to do chemistry so well that it became obvious that the smallest chunks that matter can be split into seemed to be the atoms of the periodic table. A hundred years later we realized that atoms can be split even further into nuclei and electrons. What didn't change was this idea that each chunk had its own independent existence.

This idea ran into a deep crisis during the early 20th century when we discovered the first effects of quantum mechanics. It turns out that atoms and nuclei and electrons do not, at all, behave like really small pieces of ordinary matter. Instead, they are behaving radically different, so different, indeed, that the human imagination has a hard time keeping up with their dynamic properties.

For a while we were in a limbo regarding our description of nature at the microscopic scale. It seemed like we could cling to some sort of "little weird billiard ball with mass, charge, spin etc. properties" kind of theory for electrons, but as time went by, this became ever more hopeless. Eventually we discovered quantum field theory, which does away with the particle description completely, and with that all the ontological problems of the past century have disappeared.

So what's the new way of describing nature? It is a field description, which assumes that the universe is permeated by ONE quantum field (you can split it up into multiple components, if you like). This quantum field has local properties that are described by quantum numbers like charge. This one quantum field is subject to a quantum mechanical equation of motion which assures that some properties like charge, spin, angular momentum etc. can only be changing in integer (or half integer) quantities (in case of charge it's actually in quantities of 1/3 and 2/3 but that's a historical artifact). Moreover, this field obeys symmetry rules that leave the total sum of some of these quantities unchanged or nearly unchanged. Charges in particular can only be created on this field in pairs such that the total charge remains zero.

So now we can answer your question in the language of the quantum field: the electron gets its charge by the field allowing to create one positive charge state and one negative charge state at the same time, leaving its total charge zero. This process takes some energy, in case of the electron-positron pair a little over 1MeV. Every other property that is needed to uniquely characterize an electron is created in a similar way and at the same time. The elementary particle zoo is therefore nothing but the list of possible combinations of quantum numbers of the quantum field. If it's not on the list, nature won't make it (at least not in form of a real particle state). Our list is, of course, at best partial. There are plenty of reasons to believe that there are combinations of quantum numbers out there that we have not observed, yet, but which are still allowed.

Ill answer the second part of your question about effective mass and quasiparticles, since I see that CuriousOne has answered the rest better than I could have.

In a metal or semiconductor, the electron is not in the same free state it would be in a vacuum. It is bound to (although delocalised within) a lattice of positive ions. So its dispersion relationship is different from the dispersion relationship it would have were it in freespace.

It can however move almost freely within the lattice, therefore, over length scales that are long compared with the lattice period and as long as we're not too near the edges of the lattice, its response to electric and magnetic fields is very much like what it would be to the same fields in freespace, but, because there is a different dispersion relationship owing to the lattice's presence, it behaves as though it has a very different effective mass (i.e. its acceleration in the presence of electric fields is $q\,\vec{E}/m_{eff}$ and that in the presence of magnetic fields is $q\,\vec{v}\times \vec{B}/m_{eff}$), where $m_{eff}$ is different from the electron's freespace rest mass. Depending on the band structure, the effective mass can even become negative (i.e. responding to electromagnetic fields in the opposite sense to normal). This word "effective mass" here characterises the electrons response to fields; it is not the rest mass that characterises the electron's low speed energy content (i.e. it is not the $m$ in $E^2-p^2 c^2 = m^2 c^4$).

I'm not a semiconductor / electron specialist, but the word "quasiparticle" is used in two different ways that I am aware of. The first arises when one quantizes the theory of acoustic vibrations, and the phonon is to this quantized mechanical theory of lattice vibrations what the photon is to the quantized electromagnetic field ("vibration"). The second usage is probably more relevant to the electron in a lattice, and refers to "particles" that are quantum superpositions of some fundamental free particle and a state where it is bound to, absorbed by or interactive with something. This would seem to make sense in this context of the electron in the lattice. In dielectric materials or plasmas or other materials, for example, one does not have pure photons and pure light, one has instead quantum superpositions of free photons and excited dielectric matter states. It is therefore more correct to call the quantum of this disturbance a "quasiparticle", and in the case of the photon, one calls it variously a polariton, exiton or plasmon, depending on the nature of the quantum states in superposition with the free photon.