You want to make your robots mouth move in various ways as your robot is speaking.
Your simplest movement is to only have the lower jaw on a linkage to move up and down in sync with whatever soundtrack or live audio is coming from your robot.
You could add some simple lip movements along with perhaps flexible skin around the cheeks to move as your robot talks to indicate laughter or anger or the robot giving a sneering gesture like Elvis was well known for.
These movements need linkages and some motors can perform one movement or several if you design the linkages well.
Will Cogley of Niheim Mechatronics has some great Youtube videos of his eyeball and in this case mouth mechanisms. https://www.youtube.com/watch?v=mEAz-72ZjKE
Here is a very good link to a Youtube user Gary Willett who has been posting incredible animatronic videos since at least 2008. This video shows a realistic jaw that moves back and forth (protruding). https://www.youtube.com/watch?v=Wx2yqGKz1IM