Let me take a stab at answering this (somewhat vague) question. You said you are interested in the analytic structure of QFT. But you also mentioned the RG, which is somewhat different. I will try to address the analytic structure of QFT and then emphasize that the renormalization group can be thought of as merely a trick to improve perturbation theory.
So if one promotes all momenta in a scattering process to complex variables, then the complex function one obtains is not single valued, in other words, the amplitude lives in a multi-cover of some $C^n$. All the branch cuts must live on the real axis.
The significance of the coefficients of the brach cuts is that some "virtual particles can become on shell," or more abstractly, that some subset of the external legs can create an on-shell state with the prescribed kinematics. (These are the famous unitarity cuts.)
There could be poles on the real axis too, but these should be thought of as a special case of a branch cut, one where the discontinuity is a delta function. This means that the state in the Hilbert space is “isolated” and it is thus interpreted as a bound state.
RG is quite a different set of ideas altogether. Perhaps it is best to illustrate why we need the equations of RG by an example. Consider some two point function of currents in a QCD-like theory. One removes the divergences with dimensional regularization and so the answer should be $\sim g^2$, where $g$ is the gauge coupling. However, since there are virtual particles in the loop this cannot be the right answer because there must be a branch cut. And indeed, one finds $g^2\log(p^2/\mu^2)$ where $\mu$ is an arbitrary fixed scale and $p$ is the momentum.
The branch cut is independent of $\mu$ because the imaginary part must be finite, scheme independent.
A result like $g^2\log(p^2/\mu^2)$ is highly unsatisfying practically because the expansion in $g$ breaks down at large enough momentum. This is where the RG equations come in. Basically to understand the large $p^2$ regime (which is ought to be weakly coupled in AF theories!!) one needs to re-sum perturbation theory, but one needs to worry only about the leading large logarithms which endanger the $g$ expansion. One finds that the answer can be written as $g^2(p^2)$ and $g^2(p^2)$ satisfies the RG equation, which, in QCD, guarantees that, at large $p$, $g^2(p^2)$ is small. So perturbation theory is saved. Another way to think about RG is that there is a family of possible regularizations of the theory, parameterized by some scale $\mu$, and we can compute any process we want with any choice of $\mu$ , these are all equivalent, but to change $\mu$ to $\mu'$ one needs to redefine the coupling constants. And indeed, to avoid large logarithms it is best to choose $\mu$ around the momentum scale. This is why $g^2(p^2)$ effectively becomes $g^2(\mu)$.
So to summarize, the RG's claim to fame is that it re-sums the dominant terms in perturbation theory to all orders in theories where ordinary perturbation theory is jeopardized by the presence of parametrically large pre-factors in front of the coupling constants. The presence of these large pre-factors can be understood from the presence of branch cuts in momentum space.