Keep your eye on Jupiter's moons. You'll see some neat visual phenomena: occultations, eclipses, and transits of the moons and their shadows.
An occultation occurs when Jupiter passes in front of one of its moons and blocks the view from Earth.
An eclipse is when a moon passes into the shadow cast by Jupiter.
But, the most fascinating to watch are transits. There are two main types. A moon transit takes place when one of the moons passes in front of the disc of Jupiter. It is fairly easy to see a moon near the limb of the planet just when it is entering or leaving the disk, but following its complete march across Jupiter's surface is challenging. Often, you'll lose the moon as it blends in with the background of Jupiter's atmosphere.
Even more interesting is when a moon casts its shadow on Jupiter. This is a shadow transit. The shadow is deep black and resembles a tiny ink dot that stands out starkly against the planet's colorful atmosphere. Look for the moon to follow its shadow. Quite often you can see the pair crossing the disk at the same time.