What is the difference between a pillar (rukn) and an obligation (wajib) in prayer, and when are the transitionary takbirs and the tasmi' (saying "Sami'a Allahu liman hamidah") uttered?

The questioner did not specify the paragraph that was unclear to him. However, if he meant the sum of what was mentioned before his question, then these are the duties of prayer.

The difference between a pillar (rukn) and a duty (wajib) in prayer: - The pillar: If the worshiper omits it unintentionally (out of forgetfulness) and then remembers it, he must perform it and prostrate out of forgetfulness (sujood al-sahw). If he omits it intentionally, his prayer is invalidated. If he cannot make it up, he cancels the raka'ah in which the pillar was missing and performs another raka'ah in its place. - The duty: If the worshiper omits it unintentionally (out of forgetfulness), prostrating out of forgetfulness is sufficient for him, and he is not required to perform it if its place has passed. If he omits it intentionally, his prayer is invalidated.

As for the supererogatory acts (sunan) of prayer, they are everything other than the pillars and duties.

The takbir (saying "Allahu Akbar") for transitioning between the pillars of prayer should be between the two pillars, which is preferable. As for the tasmi' (saying "Sami'a Allahu liman hamidah"), it is performed between the beginning of rising from rukoo' (bowing) and the straightening up into the standing position from it.