When you move a drawing next frame it is going to be in a different place. You can see the drawing changing places when you play the animation. But when you move an object it stays in that place the entire animation. I want to be able to have the object move and change position multiple times during the animation (every frame it will move a little). How do I do that?
While in Grease pencil draw mode when you edit over frames a new keyframe is automatically created, 3D objects (like your text object) have to explicit the keyframe insertion. This can be done pressing I to insert a keyframe, or, automatically, enabling the little dot icon (record auto keying) on the left of the centered play buttons in the timeline window (at the very bottom of your screenshot).