The rule I'm used to from previous editions is that a character with a companion such as a familiar must choose whether to use their Move and Action themselves or have their summoned creature use it. From the answer here (What can a familiar actually do?) I'm getting the sense that the familiar has an action of its own. Is this the case?
From the Find Familiar entry:
In combat, it rolls its own initiative and acts on its own turn. A familiar can’t attack, but it can take other actions as normal.
A familiar is an independent creature with its own initiative, movement, actions, and so on. So it can take an action, a bonus action, and a reaction. (Assuming it has something to do with them.)