Here are descriptions of things easily seen through most small telescopes. Except for the Moon, objects seen visually through a telescope will not look like the long-exposure photographs seen through the Hubble Space Telescope or even the amateur photos on the TAS homepage. A camera is able to absorb far more light and color over a longer period of time than the human eye. Non-planetary visual objects tend to have little to no color; some will only appear as "smudges" at first. You will have to train your eye to look for the subtle details in diffuse objects like galaxies and nebulae.
The free monthly sky chart available at Skymaps.com a great place to find items to seek out with your telescope. These maps show what constellations, planets, and deep sky objects are can be seen each month with the nakeed eye, binoculars, or a telescope. Here are some other things you can look for pretty easily, because they are visible also to the naked eye and will be easier to find. Amateur astronomers normally use binoculars to pick out their targets first, then use the telescope to go in for a closer look. Any binoculars you already own can be used!
Easier to Find
The Moon: The Moon looks fabulous in a telescope! The best time to look is before or after a full moon, when the shadow line (the terminator) crosses the face of the Moon. The low angle of the sun will reveal a wealth of features to explore.
Planets: Planets move against the backround of the stars (the name "planet" comes from a Greek word meaning "wanderer",) so you will to find out which constellation has a planet in it at the moment, and you'll need to know how to find that constellation. But once you do, your telescope should clearly show Saturn's rings and Jupiter's four Galilean moons as well as cloud bands. At the right time, and with a little luck, you may be able to see polar ice caps on Mars and a little bit of surface detail. You will be able to watch Venus go through her phases (just like the moon).
Star clusters: These groups of gravitationally-bound stars spilling across the night sky in your eyepiece have been compared to diamonds spilling on black velvet. In Fall and Winter, look for M45 (the Pleaides), which is a naked eye cluster in the constellation Taurus, the Bull.
Galaxies: The Andromeda Galaxy, in the constellation Andromeda, is visible to the naked under as a hazy patch, under very dark skies. It will fill the eyepiece of your telescope. Look for an elongated smear of light. The light hitting your eye is from another galaxy! How cool is that?
More Challenging to Find
Here are some objects that will require more skill (and probably binoculars) to find. The are not visible to the naked eye, so you'll need to know a little about the night sky and be able to follow a star chart or planetarium app. If you decide to stick with astronomy, you might invest in a more detailed star atlas, such as the Sky & Telescope Pocket Sky Atlas.
Planets: Uranus is actually visible in a telescope, but you have to know exactly where to look. Use the internet or a planetarium app to determine where Uranus is right now. You are looking for a blueish "star" against the background of the other stars, that is not on any of the star charts. That's about all the detail you can see, but most people have never seen Uranus at all! It's worth a try. Don't bother looking for Neptune -- it is beyond the reach of this small telescope.
Asteroids: Yes, you can see asteroids and minor planets with this scope, if you know where to look! Check a website or planetarium app for the positions of the minor planets Ceres and Vesta. As with Uranus and Neptune, these will just appear as "star like points" and you'll have to find them based on the "which one of these doesn't belong" on the star charts method. This is exactly how these asteroids were originally discovered, so you'll be following in the footsteps of our astronomer ancestors!
Star Clusters, Galaxies, and other Deep Sky Objects: All of the 110 Messier Objects are within reach of this telescope. You'll need a good set of star charts or app, but these are well worth seeking out. Charles Messier's Eighteenth Century "List of things to Avoid Looking at Because They Are Not Comets" has turned out to be the treasure box of amateur astronomy. His objects include the most beautiful and interesting objects easily visible in a small telescope. They include galaxies, nebulae, and star clusters all across the northern hemisphere (basically everything visible from the latitude of Paris, where Messier's observatory was located.) Here are just a few of the ones that will be visible in the Library telescope (if you can find them!)
M44 The Beehive Cluster in Cancer
M36, M37, and M38 in Auriga
M21, M22, and M23 in Sagittarius
M13 a Globular Cluster in Hercules
M42 the Orion Nebula in Orion
M41 open cluster in Canis Major