How does an electron move through a metal or a semiconductor?

You do not need quantum electrodynamics. The usual way in which conduction is understood goes by the name of linear response theory and in more specific contexts it is referred to as the Kubo formula. See the wiki page for Kubo formula. There are more exotic ways of calculating conductivity but these are used in specific contexts when the Kubo formula fails.

The idea behind the Kubo formula is straightforward even though the math surrounding it can get ugly. You start with a Hamiltonian $H_0$ for the system in question. You can think of $H_0$ as describing a metal. You want to measure the current $j\sim\langle \hat{j}\rangle$ where $\hat{j}$ is the current operator and the brackets denote taking a thermal expectation value. If I do not attach a battery to my metal there will not be a current. In other words if I take the expectation value using the system described $H_0$ then there is no current.

To get a current I have to add a term $V$ to the Hamiltonian and take the expectation value with respect to the Hamiltonian $H=H_0+V$. This added term simply describes the coupling of the original system to a classical EM field. In some cases it is important to consider the EM field as a quantum object but in many cases classical EM suffices. Since it is difficult to diagonalize the combined Hamiltonian $H=H_0+V$ what people do is treat $V$ using first order perturbation theory, hence the name linear response. Since linear response is just first order perturbation theory the current that you calculate will depend on the eigenstates of $H_0$. Knowing these eigenstates gives you insight into the conductivity of a system.

To expand the previous answer.

You can model conduction properties with the Kubo formalism, but also using the Landauer-Büttiker formalism. It basically states that every conduction problem can be considered as a scattering problem. A key difference between Kubo and Landauer is that the first one is time-dependent problem, while the second is not. If you want to dig for some details, I think the books by Datta are more or less standard. I like pretty much the discussion by Di Ventra: Electrical Transport in Nanoscale Systems, and it covers both formalisms.