If you are observing with a Newtonian reflector, the image will be rotated 180 degrees (generally south is up). In a refractor or Cassegrain with a mirror diagonal the view will be mirror reversed (north up, but mirror reversed). See this explanation of directions in the telescope.
Jupiter
A complete novice can expect to see two main cloud bands on Jupiter and its four Galilean moons. With more practice, not only the South and North Equatorial Belts (SEB and NEB), but temperate belts in each hemisphere may also sometimes come into view, as well as darkened polar areas.Above: The moon Io and its shadow visible against the cloud tops of Jupiter. Image by Steve Hill, CC by 2.0, via Flickr
Below: The four Galilean moons are aligned on one side of Jupiter in this image by Ivana Peranic, CC by 2.0, via Jeremy Keith/Flickr.
Jupiter's Galilean moons—those that Galileo was able to see in his tiny refractor: Ganymede, Callisto, Io, and Europa—are the only moons, out of the currently identified 95 Jovian moons, that are visible to amateur observers, and can even be spotted in binoculars. Because their orbits are well known, predictions as to transits across the face of the planet and the corresponding shadows, disappearances and reappearances behind the planet or its shadow, and even occasional occultations and eclipses of one moon by another are available. You can plan an observing session to add these to the interesting details you can see in your telescope.The easiest are the shadow transits, which show up as dark black dots on the face of Jupiter. The moons themselves are more difficult to see when they pass in front of the planet, and much depends on the level of contrast with the cloud deck below them. I have seen them many times in my 4.5-inch reflector, but have been unable to see them just as many times.
Averted vision is unnecessary for Jupiter and its moons. In fact, you'll see the most by looking directly at any feature. Bore your vision into the feature, almost as if you are looking through it, to get the most detail to register. Relax your eye and just let the detail burn into your retina. Really stare into it!
Sketching the cloud belts and swirls that you see can really help you focus on the details. You don't always have to sketch what you see, but try it a few times and you'll be surprised at the amount of detail that is actually visible. You may not see it all at the same time, the same with deep sky observing, but you will build up a complete picture with fragmented glimpses. This teaches you to place a detail within the greater context and you'll also see how the features slowly traverse the globe of the planet in an (astronomical) westward direction as Jupiter completes a full rotation in less than 10 hours—the fastest rotating planet in the solar system. For more on observing Jupiter, I recommend How to Observe Jupiter Through a Telescope by BBC Sky At Night Magazine.
Saturn
Of all the sights a beginner can see in the telescope, Saturn is probably the most striking. When I show it to people at public outreach events, most people are thrilled and some even question whether what they are seeing is real.While Saturn doesn't show nearly the same amount of detail as Jupiter, and it's remarkably smaller in the eyepiece, the sheer beauty and uniqueness of the planet will keep you coming back whenever you can. Something about the rings is precious. Really.
Above: Saturn by John Spade, CC by 2.0, via Flickr
The rings change their tilt over the years, and with Saturn now in the evening sky, the rings are nearly edge-on. This makes it difficult to see the major feature in the rings, the Cassini Division. This thin dark lane is sometimes visible on nights of excellent seeing with the rings tilted towards or away from us at a significant angle. Look for it at the outward ends of the rings, where they become more visible because they begin to curve the other direction and the gap is seen at its fullest width. This gap that appears so tiny to us is actually almost 3,000 miles wide! The next ring plane crossing is in March 2025, when the rings, being an average of only about 30 feet thick, become invisible in our telescopes. The Cassini Division may have to wait.
Above: Saturn's varying ring tilt, image by NASA and the Hubble Heritage Team (STScI/AURA), CC by 2.0, via Flickr. Cassini Division label added.
If you look carefully you will usually see a slightly darker band around Saturn and perhaps some subtle shading elsewhere, especially at the poles. Saturn is much smoother than Jupiter, but it does have very infrequent storms visible in our telescopes, such as the great white spot of 2011.
For Saturn's moons, you'll have to use averted vision for all except the largest, Titan, and Iapetus when it is furthest out on the western side of Saturn and its bright icy side is turned toward Earth. Iapetus strays pretty far from Saturn in its wide orbit and can easily be confused with background stars. The inner moons are dimmer, but with good seeing, patience, and a telescope of around 4 inches or more, you should be able to pick out Rhea, Tethys, Dione, and possibly Enceladus. Mimas is quite difficult, Hyperion requires a larger telescope of 10 inches or so, and you won't have a chance at any of the other moons of Saturn, which currently number 146 and counting.
Jupiter and Saturn observing resources:
Cloudy Nights Planet Gallery (more recent images at top)
Cloudy Nights Major and Minor Planetary Imaging thread (latest images)
Online interactive observing tool for Jupiter's Moons (Sky & Telescope)
Great Red Spot transit times (Sky & Telescope) (when it crosses the planet's central meridian)
Online interactive observing tool for Saturn's Moons (Sky & Telescope)
Apps: