Slang is specifically informal or colloquial speech used by certain social groups. An idiom is a phrase whose actual meaning isn't obvious from the literal meaning of the words in it. Idioms can be colloquial or slang, but they don't have to be.

For instance, English has phrasal verbs (a verb plus a preposition or adverb), which are idiomatic and in very common use: "give up", "think over", "put up with", "look up", "look after", and so on. Looking up something (as in a dictionary or catalog) does not involve any upward motion or direction. Thinking over something does not involve being spatially located above that thing.

As an example of an idiom being used in a formal, scientific setting, Okrand's Ph.D. dissertation on the Mutsun language has "Her comments about Mutsun are useful, but must be interpreted in light of the fact that Mutsun was, at best, a secondary language for her." (page 6). The underlined phrase is an idiom: the fact does not actually cast any light. The intended meaning is "while considering the fact".

The idioms in KGT and elsewhere aren't described as being colloquial or slang, so they're probably fine to use in formal situations.