[noun] + [verb] + [noun] + [verb]
[phrase] + 而 + [verb]
[verb] + 之 + [phrase]
以 + [noun] + [verb phrase]
A 成语 – a four-character idiom – preserves classical grammar. The words inside it follow rules modern Chinese has dropped, so a chengyu you can't parse often opens up once you know those rules.
Most chengyu split into two halves of two characters, and the halves usually mirror each other: 东张西望 is "look east, gaze west".
A noun before a verb can say how the action is done. 狼吞虎咽 is "swallow like a wolf, gulp like a tiger"; 蚕食 is "eat away like a silkworm".
These four function words do their classical jobs inside a chengyu. 之 is "it" as an object: 置之不理 "set it aside and ignore it". 而 links two actions or a manner and an action: 半途而废 "stop halfway". 以 is "by means of": 以身作则 "make oneself the model". 于 is "from" or "than": 青出于蓝 "blue comes from the indigo plant" – the student outdoes the teacher.
Classical Chinese put a pronoun object before its verb after a negative, and a question word before its verb. 时不我待 is "time does not wait for me": 我 is the object of 待. 何去何从 is "go where, follow where". 唯利是图 brings its object forward with 是: "profit alone is what they seek".
A chengyu is already a complete phrase. A verb-object chengyu can't take a second object, and one that describes manner doesn't take 很.
How a chengyu works as a predicate, a modifier or an adverbial has its own point.