I think collocations are much more important. They are used all the time, in both everyday and formal English. They are almost like "vocabulary," except that they are made up of several words instead of just one. I'm not sure where "phrasal verbs" leave off and "collocations" begin, but they are important to understand.

Native speakers are not taught them in school. We pick them up. We are not even conscious of using them. Collocations are something best learned from a professional teacher who teaches English as a foreign language.

Idioms are fun, but they are a little dangerous. We are entertained by colorful idioms, but many of them are listed because they are colorful and amusing. They are not used all that often. They are frequently regional. Try asking native English speakers "Have you ever heard anybody actually say 'raining cats and dogs?'" Many will say "never." Many will say "My aunt Nancy always used to say that." Many will say "Of course, I say it all the time."

Lists of idioms are unreliable. They often include regionalisms, outdated idioms that are rarely used, or colorful catchphrases that are mostly "signature phrases" connected with media personalities. (Did anybody but Red Barber ever say "he's sitting in the catbird seat?")

It is helpful to understand idioms but you almost never need to use them. They do not make it easier or more comfortable for English speakers to understand you. In fact, if you have the normal amount of accent that "fluent" foreign speakers usually have, they will sound so strange that native speakers will not be sure they have heard you correctly. 

Clear, plain, logical English is always fine. However, unlike idioms, using the right collocations does "improve" your English in the sense of making it easier and more comfortable for natives speakers to understand you.

Collocations are used in everyday speech, in informal and formal written and spoken language.
Yes, collocations are important.

Idioms are figurative speech and cannot be understood literally (the words don't have their usual meanings)
Idioms are nice to know but difficult to use, and misusing them can result in embarrassing situations.
I always recommend that English learners learn idioms only after they have reached the advanced level.