To understand the essence of perturbative renormalization you don't need any quantum field theory nor any quantum mechanics. A simple toy model suffices.
Suppose your theory makes a prediction for two distinct observables $F$ and $G$ in terms of a perturbative parameter $g$:
$$F = g + g^2 (S+1) + g^3 (S+1)^2 + g^4 (S+1)^3 + ...$$
$$G = g + g^2 (S-1) + g^3 (S-1)^2 + g^4 (S-1)^3 + ...$$
Here, $S$ represents some divergent mathematical sum. For instance, $S=1+2+3+4+...$. As in both expressions each term beyond first order in $g$ diverges, your perturbative theory seems pretty useless. What to do?
Under perturbative renormalization, one attempts to eliminate the non-observable parameter $g$ from the theory, and rewrite the perturbative series in terms of the observable $G$. So, we write:
$$g= \alpha_1 G + \alpha_2 G^2 + \alpha_3 G^3 + ...$$
and substitute this expression in the perturbation expansion for $G$. This gives us (up to order $G^3$):
$$G = \alpha_1 G + \alpha_2 G^2 + ... + (\alpha_1 G + ...)^2 (S-1)+ ...$$
It follows that $\alpha_1 =1$ and $\alpha_2 = -(S-1)$:
$$g= G - G^2 (S-1) + ...$$
Substituting this series in the expression for the observable $F$ yields:
$$F= G - G^2 (S-1) + ... + (G-...)^2 (S+1)+ ...$$
Which gives us the end result:
$$F = G + 2G^2 + ...$$
Note that this perturbative expression for $F$ is written solely in terms of the observable $G$, and no longer contains the divergent quantity $S$. You conclude that your perturbative theory for the observables $F$ and $G$ is renormalizable.
Note: when performing a perturbative renormalization and all infinities exactly cancel each other, it feels a bit like magic. In this toy example the magic is a direct consequence of the hidden non-perturbative relationship $F-G = 2FG$.