For example, I want to consider the following situation: photon transit from $m$ energy level to $m+2$ after absorption of two phonons with frequency $\Omega$. I want to calculate a transition rate for this process and now I'm little stuck in the choosing the way to do it. Should I use a second-order perturbation theory to find this rate, or can I just multiply the probabilities as a sequential processes ($m \to m+1 \to m+2$ states) to find total probability? Which of these options is more correct?
To be honest, it depends on the situation, and particularly on how long the pump lasts and what the lifetime of the states is. If you have long-lived states, then sequential processes become possible and they can dominate, but at shorter timescales you expect second-order perturbation theory to be more crucial.
It also depends on how close you are to the resonance on the middle transition. If you're exactly on resonance, then SOPT in its naive form will diverge, until you include the lifetime of the state, and you might be better off with a sequential process. On the other hand, it's quite common to have the full two-photon process be exactly on resonance, but not the middle step; in these cases SOPT is essential.