What is the technical term from a group of consecutive words with a single associated meaning? For example, phrasal verbs like: "get out" and idioms like: "on the other hand".
The short answer is that there is not a single theory-independent term. It depends on the type of relationship. So "get out" is a "phrasal verb" or "prepositional verb" and "on the other hand" is an idiom.
In construction grammar, all of them could be described as constructions but you'd still want to differentiate between them in communication.
Things are a bit easier with word groups that don't have a relationship but tend to appear together like "bread and butter". Those are called "collocations". I wonder if that's what you were driving at.