There's quite a lot to see in the vast night sky. The scale and variety of objects defies comprehension. Here I'll try and categorise everything in a way that makes knowing what to look for a bit easier.
Planets are something you're probably familiar with. These are our neighbour worlds in the solar system. Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune. Between all of these, there is a lot of variety and interesting details which set them apart.
Planets were among the very first objects to be studied with telescopes. Due to their relatively close distance to our world, they are among the few objects that appear as distinct shapes, rather then as point-like stars. This allows detailed features to be observed and studied.
Jupiter is an awesome spectacle as far as planets go. Being the largest of the planets, it can appear relatively bright, big, and full of detail. Distinctive cloud bands can be observed, as well as the famous "Great Red Spot", a gigantic storm larger then our entire planet! Jupiter's 4 largest moons can also be seen through almost any telescope.
Saturn is probably the most interesting planet to behold, due to its large, incredible ring system. Even small scopes can make out the rings. Larger scopes can make out the Cassini's Division, which divides the inner and outer rings.
Mars is one of the few rocky worlds observable through a telescope. The "red planet" is small - only 15% of Earth's volume. This means that how it appears depends on how far away it is, at any given time during our different orbits. When far away, very little can be made out. When close, however, Mars's ice caps and surface textures can be observed with notable detail.
Venus is the brightest planet in the sky, and also usually the closest to us. However, its atmosphere is much too thick to observe its surface, so details are absent. It is possible to see its phases, however, just like our moons. The other planets can be seen, but are difficult to find, and show less detail. They can be worthwhile nonetheless. You can tell if you've found Uranus or Neptune by their respective green and turquoise colors.
Star Clusters are just what they sound like. They are beautiful spectacles, which can range from small groups, to incredibly dense groups of millions of stars.
Open Clusters are loosely-bound groups of stars, often up to a few hundred. These stars tend to be young, hot blue stars (although there are exceptions). Notable examples of these are Pleiades and Hyades, both of which are easily seen by the naked eye. These star groupings are noteworthy enough to have been known since ancient times throughout the world. Open clusters show considerably more stars when viewed through a telescope, and can be quite brilliant.
Globular Clusters contain significantly more stars then open clusters - up to several million, in roughly equivalent spaces. These star groups are thus extremely densely packed. Also, globular clusters tend to be older, and contain more yellow and red stars - though again there are exceptions. Examples of globular clusters are Omega Centauri, and the Great Globular Cluster in Hercules. While quite far away, these are still visible to the naked eye, despite their tremendous distances. In telescopes, they appear as round, hazy - yet dense discs. Large aperture scopes can resolve individual stars, which can make for a truly stunning sight to see.
While star clusters are impressive, they are small spectacles compared to entire galaxies. Galaxies are enormous structures made of hundreds of billions of stars and every other type of object. A common nickname is "island universes"; which suits them well. It turns out that nearly everything you see in the night sky is within our own Milky Way Galaxy!
While the "Milky Way" is commonly associated with the beautiful, hazy band of white light arcing across the sky, this is merely a part of the galaxy closer to the center, which is further away, and appears more condensed. Every star that can be seen in the sky is part of the Milky Way, and we happen to be within one of the spiral arms.
Other galaxies can be seen in the sky. Among the most notable are the Andromeda Galaxy, Triangulum Galaxy, and large Magellanic Clouds. These galaxies are far, far more distant then any other object that can be observed. As a matter of fact, they are the furthest objects that can be seen with the naked eye in the right conditions.
The Andromeda Galaxy is estimated to be roughly the same size as our Milky Way. It can be seen as a hazy patch in dark skies without a telescope. It is the closest large galaxy to our own, and features the classic spiral shape most people associate with galaxies. With a telescope, the structure and some detail can be observed. The Triangulum Galaxy is similar, yet smaller and slightly further away.
The Magellanic Clouds are a pair of comparatively smaller, irregularly-shaped galaxies, known as dwarf galaxies. They are both close to (and perhaps orbit) the Milky Way. These galaxies are only viewable from the southern hemisphere. They appear like separated pieces of the Milky Way to the naked eye.
A nebula is an interstellar cloud of dust, hydrogen, helium and other gases. They are diffuse, containing little or no well-defined boundaries. They tend to be quite large, reaching up to hundreds of light years across. A noteworthy example is the Orion Nebula. It is one of the brightest nebulae, and is visible to the naked eye. It is also one of the most scrutinized and photographed objects in the night sky.
Different types of nebula exist. While many are simply large interstellar clouds, some are actually leftover remains of stars. Some, which have shed their outer layers, are known as "Planetary Nebula". An example is the Helix Nebula, which has a large ring of matter surrounding a small white star. Very large stars, which end up exploding as supernovas (some of the largest explosions), leave supernova remnants - vast, superheated clouds of exploded star material. An exampled is the Crab Neblula, which is the remains of a supernova that occurred in the year 1054, and was briefly visible from Earth.
Although nebula are outstanding objects to see through a telescope, keep in mind that the pictures you often see of them come from highly specialized telescopes. These are much more sensitive to color and fine detail then our own eyes are. It turns out that most nebula appear mainly black and white through our own telescopes, with a few mild exceptions. They are nonetheless impressive, just so long as you don't expect to see exactly what's printed online or in a fancy magazine.
Nothing beats the moon when it comes to astronomy, due to being the brightest object in the night sky, and the closest object to our own world. It is by far the best object for observing fine detail and surface features, and through a telescope, these are truly amazing sights.
Among the numerous features the moon has to offer, are countless impact craters, several noteworthy mountain ranges, and large, dark basins created by ancient lava flows, called Maria. Of the craters, Tycho, Copernicus, and Kepler are especially noteworthy, as each displays a dramatic pattern of long rays radiating outward.
Although you might think the best time to observe the moon is when it's full; this is actually not the case. Since the sun is shining directly down onto the surface, there are virtually no shadows to provide surface details or relief. A full moon is also so bright that it can be uncomfortable to look at for long, especially through a telescope. The best time to observe would be anywhere between a crescent and a little over half full. A moon-filter, which screws onto a telescope eyepiece, can help also cut the glare significantly.