As defined here, a small telescope is a refractor (a telescope with a big lens in the front of the tube, and an eyepiece holder at the back) with a lens diameter (D) or aperture between 50mm (~2 inches) and 90mm (~3.5 inches). Any refractor smaller than 50mm will be practically useless for astronomy, and any larger than 90mm is too large (and expensive) to qualify as a "small" telescope. Newtonian reflector telescopes (with a mirror at the back of the tube and the eyepiece holder near the front) between 3 inches and 4.5 inches will probably fall into the small telescope category as well.
In the eyes of overly serious amateurs, most inexpensive telescopes in these size ranges are useless. It is true that they have their shortcomings, but it is possible to do astronomy with these telescopes. If a small telescope was all you could afford, and/or you can't return your purchase, read on.
ESSENTIALS
There are some features a telescope needs in order to be useful for viewing astronomical objects.
Any telescope can be made to magnify any amount, simply by changing the eyepiece. In practice, the high powers advertised with many small telescopes will be unusable. The eyepiece determines the magnification, so here is the scoop on eyepieces.
Eyepieces are marked by their focal lengths. The longer the focal length of the eyepiece, the lower the magnification when used with a given telescope. The objective lens or mirror also has a characteristic focal length. The longer the focal length of the objective, the higher the magnification when used with a given eyepiece. You can find out the magnification (or power) a telescope will give by dividing the focal length of the objective (often marked on the box as F=700mm or something similar) by the focal length of the eyepiece.
For example, I use 4 eyepieces with my 60 millimeter refractor. The focal length of the objective is F=710mm. The focal lengths of the eyepieces are 20mm, 12.5mm, 9mm, and 6mm. The 20mm gives a magnification of 710 / 20 = 35.5x. This is a good low power for finding and viewing most objects.
Make sure you have a low power eyepiece in the range of 20x-40x. If you don't, you will have a tough time finding anything! The lower the power, the wider the field of view and the easier it is to locate objects. When you have a small, bright object like a planet in the field of view, you may wish to zoom in on it. This is where a higher power eyepiece will come into play. My 9mm eyepiece gives a magnification of 710 / 9 = ~79x.
Ridiculous eyepieces with focal lengths of less than 6mm are sometimes shipped with small telescopes. These are, without exception, poorly made and useless. They serve only to allow the manufacturer to advertise a high magnification. In practice, the view through a 4mm eyepiece will be uncomfortable, dim, and blurry. Just as useless is the Barlow lens, which effectively increases the magnification between 2x and 3x. A well-made Barlow might work, but the ones shipped with small telescopes have poorly-made lenses and sometimes will not even focus!
THIS IS TOO COMPLICATED! I JUST WANT TO LOOK AT THE SKY WITH MY TELESCOPE!!!!!!!!!!!!!!!!
OK. This is actually a good time for it, if it's clear out and the Moon is up. Bundle up and take your scope outside. Put in your low power eyepiece and point the scope at the Moon. Center the Moon in the finder. Look in the eyepiece of the main scope. If you don't see the Moon, your finder isn't aligned. Luckily, the Moon is so bright that you don't need a finder. Just keep looking in the eyepiece and sweep the scope around the area until the Moon comes into view.
When you come across the Moon, it will probably be out of focus. So, find the focus knob and twist it until the Moon comes into focus. Did the scope move much when you did this? If so, your mounting is not the steadiest. You can either try to fix this or just live with it. What about the Moon's image? Can you see craters? The best part of the Moon to look at is the terminator, the line separating the sunlit side from the night side of the Moon. Shadows are long here, so detail will be enhanced. You should be able to get a very sharp image, with lots of craters and bright spots and dark spots visible. The Moon will be the most detailed astronomical object you will ever see in your telescope, so if it doesn't look good, then nothing else will. If something doesn't seem right, you might want to find someone to evaluate your telescope's optics.
Take some time to look at the Moon. You'll notice that the Moon appears to move out of the field of view. This is due to the Earth's motion, so you'll have to push the telescope every couple of minutes to keep up. Put in a higher power eyepiece and refocus. You may not be able to fit the whole Moon in the eyepiece field, now that the image is larger. Details are larger as well, but on the whole the image doesn't look as sharp. That's the trade-off of high power. You'll want to try different powers to find the eyepiece that gives the best view. Before you leave the Moon, adjust the finder scope so the Moon is in its center when it is centered in your main scope. This may take a bit of work, but it will help when you are looking for fainter objects.
Do you want to look at a planet? Do you know which planets are out now, and where to find them? If so, turn your scope on them. If not, you'll have to put off this step until you find out. Monthly astronomy magazines give the locations of the planets, usually on easy-to-read sky maps. In the meantime, let's look at a star. Any star will do, but you might want to choose a bright one. Center it in your finder, and it should be in the field of view of your low power eyepiece. If you were using a different power on the Moon, you may need to re-focus. The star will be in focus when it is smallest and sharpest. In fact, it should look just like a very bright point of light. Stars are so far away that they will always appear as tiny points when in focus. This doesn't mean that they are uninteresting. With your naked eye, you will notice that some stars are colored. You can study these colors more carefully under the telescope. In addition, some stars are double. Here are a few double stars that are good in small telescopes. You'll see that knowing the constellations and being able to read a star chart will help you out.
THE PLANETS
Small refractors are often said to be good for the moon and planets, if nothing else. The planets will usually not be as detailed as they are through a larger telescope, but they are still worth looking at. In fact, most of the descriptions of planets in SKYTOUR are valid for small telescopes. You will be able to see the rings of Saturn and the moons of Jupiter and the phases of Venus when these planets are favorably placed in the sky. Because images in small telescopes are dim at high power, you will have to use lower powers than you would in a larger telescope. This can make the planets' disks appear disappointingly small. However, if you stick to powers in the range of 80x-120x, you should be able to make out increasing amounts of detail as your eyes become trained.
THE NEXT STEP: DEEP SPACE
If small telescopes are regarded as good planetary performers, they are usually written off for deep-sky objects. Galaxies, nebulae, and star clusters are very faint compared to the Moon and planets. The larger a telescope's aperture, the fainter the objects it can see. Telescopes with apertures of over 6 inches have become standard for deep-sky work. It is true that larger scopes will allow you to see fainter objects--there is no getting around that. In addition, more detail will be observed in bright objects with larger scopes. However, the brighter deep-sky objects are also visible in small telescopes, and there is no reason you should not observe them if you want to. First, you will need to have a good knowledge of the sky. There are many books on observing the sky with binoculars and small telescopes; many of them have star charts with the brightest deep-sky objects labeled. You should also get a planisphere, so you know what can be viewed when. Objects like the Orion Nebula (M42), Hercules Cluster (M13), and Andromeda Galaxy (M31) may even be labeled on the planisphere. Perhaps the best objects to view with small telescopes are open star clusters.
It is true that small telescopes are not the easiest (or even the most cost-effective, when purchased new) way to view the cosmos. Still, if you have one you can get a lot of use out of it and learn a lot about the sky before you graduate to a larger scope.