Dressing vs. renormalization

A thorough course in renormalisation (like in Collins' book) is really what is necessary to address your concerns, as one should appreciate the big picture of renormalisation as a whole.

Nevertheless, in a nutshell, when we have a free theory, and we add interaction terms, then all of the Green's functions will acquire corrections, as will the parameters of the theory.

The 'dressed' quantity means that the fact it is an interacting theory has been accounted for and quantum corrections are included. For the propagator, we would have,

$$\frac{i}{\gamma^\mu p_\mu - m} \to \frac{i}{\gamma^\mu p_\mu - m -\Sigma(p)}$$

where $\Sigma(p)$ is an infinite sum of a certain class of diagrams. Notice this also has the effect of changing what the physical mass is, since the pole of the propagator has changed.

Renormalisation is a scheme to remove divergences from our computations. Doing so requires adding additional diagrams with counterterms. To generate these involves re-writing the Lagrangian in terms of 'renormalised' quantities plus counterterms.

Note $\mathcal L$ as a whole is exactly the same; it's a change of notation at this point, but it makes a difference in that we treat the entire counterterm Lagrangian, even the mass term, as being an interaction term instead.

In doing so however, the relation between the renormalised parameters and the bare parameters involves a subtraction point or scale $\mu$. The renormalisation group addresses then how the parameters of the theory change with this scale.