When we consider ordinary phenomena like the attraction of a proton and an electron, we usually use electrostatics and thus Lorentz’ law. The same holds for magnetic phenomena like induction or the force of two wires with a certain current. Then, we can use the non-relativistic magnetostatic theory to describe forces and motions. But of course things change if we have a) a combination of electric and magnetic phenomena – electrodynamics or b) a situation in which bodies are moving at relativistic speeds.
Then, we are in a regime where we have to use relativistic electrodynamics. Most students fear the unification of electric and magnetic field to a relativistic theory. Nevertheless it is necessary – electric and magnetic field are basically the same entity! So to understand electrodynamics, we have to understand it also from a relativistic point of view. Otherwise we will never be able to admire the beauty of electromagnetic waves, antennas and much more related phenomena.
So, how can we get to understand relativistic electrodynamics? I think the best way is to first familiarize with the mathematical background, i.e. Lorentz transformations and tensorial analysis. Since all relativistic physics is written in this language, it might actually not be a bad idea to come into contact… Then, we may check that the Maxwell equations are indeed invariant under such transformations which gives us a feeling of how to really use the theory. Then, of course, it is very important to calculate certain well-known problems. In this way we get a further feeling when we have to use relativistic electrodynamics, since we will often make series expansions that will allow us to find when it is necessary to use the full relativistic theory. This approach can be used later on for example in Post-Newtonian gravitation, an approach to expand Einstein’s theory of gravitation into powers of v/c.
You can see there is a lot of physical insight and benefit in understanding a fully relativistic theory of electrodynamics! I hope you can will be able to use it for your advantage!