As it was explained to me, we must master the coarse before we can master the subtle, and we must master the subtle before we can attain first-hand direct vision of Dharma.
In this vein, during a single meditation session we must first work with the body, then work with the feelings, then work with the mind, and only then the insight meditation can happen naturally.
So first as we sit down our body is usually disturbed and tense. The goal at this phase is to relax and calm down the body so it can sit comfortably (as opposed to forcing it to sit still).
As the body physically calms down to a sufficient degree, we should start paying attention to the breathing. The breathing can be noisy or stuck or gasping etc. Our goal at this phase is to calm down further, using the breathing as our feedback indicator.
As we work with the breathing, we may start noticing that the problem is really with the emotions. At this point we should scan the area around the body for psychosomatic projections of emotions (tense or painful feelings in chest, abdomen, veins, forehead etc.). We should identify the blockages, etc. and release them by letting go of the subconscious will to sustain them.
We proceed in this manner, as to get very comfortable. Our goal at this phase is to make ourselves feel very good, what they call "rapture". To achieve that we can employ a combination of letting go of bodily and emotional blockages, as well as intentionally generating a joyful mood.
"Rapture" is the cruder kind of good mood, while "bliss" is a deeper and more subtle variety of the same. Our goal is to generate rapture first, then let it deepen and stabilize to become bliss. At this point our attention should include physical, breathing, emotions, and subtler psychosomatic projections, all joined into one "energy map". Our job is to fabricate deliberate effort in order to purify this whole thing, and make it transparent and energized.
As we succeed in the above, we can notice that the quality of our mind is also connected with the overall energy structure. If the mind is shaky or jumpy or viscous or sleepy etc. it will affect our ability to fine-tune the system into its perfect state. So we must condition the mind to get into a perfectly open/transparent/upright state, just like we did with our body, emotions, and psychosomatic energy before.
By performing this practice again and again, first not so well, and then better and better, we can't help but get plenty of insights into the way the mind functions. As we start getting those insights, many of Buddhist theories we've read before will stop being mere theories and will become first-hand experience. This is the phase when we will find ourselves contemplating suffering, cessation, impermanence etc.
Now if you re-read the Anapanasati Sutta with the above commentary in mind, it should hopefully make more sense.