You're probably noticed that some verbs (and some phrasal verbs) need two people for the action. If only one person is the subject, then the other person must be added using "with". Who do you get on with best? Do they get on well? They try to get on. In the last two examples, it is possible to add "with each other". I think 郭东兴 has made a fair point - the meaning can change if the context changes. Notice that the thing which follows "with" in his examples is a duty, obligation or task.

"Who do you get on with best" is correct. It means who do you get along with best. I hope this helps :)

it can be used as a stand alone phrase "get on with it!" to express impatience. It can also be used to begin an activity: "Now that everyone knows what to do, let's get on with painting this house." This usage also expresses some impatience to a lesser degree The first type is much more common. The phrase is generally considered rude, regardless of how it's used.