Graphics should move on the X (horizontal) and Y (vertical) axis primarily, and with restraint on the Z-axis. Be careful with diagonal moves and avoid spinning or swiveling objects if not essential to the story.
Above all, all movements should be carefully animated with dynamic ease-in and ease-out. Usually, default ease-curves need to be tweaked to get enough energy in the animation. Never ever animate linear motion that starts or stops without acceleration/deceleration. Subtle anticipation, overshoot and secondary motion should be used to breathe some life and energy into the animation (and avoid a power-point animation look).
If there is an audio track (music and/or voiceover), great attention should be taken to time the animation to the audio in order to enforce the message.
When animating objects on or off, make sure to use a few consistent principles throughout the production. Avoid crazy text reveals, often as simple fade in or type on will do the job. Consider using animated objects (for instance images or background plates) as image masks to reveal type and icons – simple yet very effective.