Microscopy is the hobby/science of magnifying small things to view them in greater detail. It is not terribly expensive to buy a microscope and the ability to view microscopic creatures and objects is simply amazing.
There are several types of microscopes. The simplest (and least expensive) are optical microscopes. These utilize magnifying lenses of various strengths, although the most common are x10, x48, x72, and x100. Most optical microscopes use an objective lens combined with an additional lens in the eyepiece to achieve magnification. On the better models, there is a mechanical stage that holds the slide being viewed and this stage can move up or down to focus the image. A light situated below the stage provides light through a diaphragm, or hole, in the stage.
Other types of microscopes include the scanning electron microscope and scanning tunneling microscope, which are very expensive and usually owned only by businesses and universities. Both of those models use electrons to form an image, which is of much higher resolution and magnification than standard light microscopes. The STM microscope can also be used to move atoms and molecules in high precision experiments.
To get started with microscopy, you need to find a microscope that you can use. If you are in school, you can probably find one there in the biology department. If you have some money and want to buy your own, you can get a very nice model for $100-200. Avoid plastic models or models that project the images onto a screen. A good brand is national optics and can be found online for very reasonable prices.
You are also going to need some other equipment. You will need some glass slides and glass slip covers. Avoid plastics since they scratch easily and don't get as good suction. Other optional items include an eyedropper, rubbing alcohol, prepared slides, iodine, and slide cement. Eyedroppers are good for depositing aquatic specimens on the slide without contaminating the sample. Rubbing alcohol is used for sterilizing and cleaning slides before and after use. Prepared slides are glass slides with specimens permanently stored on them for easy viewing. Iodine is used to dye slides to make viewing of certain specimens less difficult. Lastly, slide cement can be used to make permanent slides.
Once you have all of this equipment, you need to set up the microscope. You want the microscope on a level surface and the mechanical stage should be as low as possible. You will also want to make sure the lowest power objective lens (usually the shortest) is the one in use. The microscope should always be setup this way to avoid scratching the high power objectives, a common costly error.
You now need a specimen. Good specimens to start out with include pond water, which contains a lot of microorganisms, and objects such as or paper, leaves, or ants (make sure you kill it before putting it on the slide).
Take a glass slide and clean it, making sure to rinse it with water if you apply alcohol or any solvents. Dry it off completely and set it down. Next, take your specimen and carefully deposit it in the middle of the surface of the glass slide. If your specimen is dry, you will need to place a small drop of water on the specimen. Once this is accomplished, take the glass cover slip and carefully place it directly onto the specimen, pressing down until it makes contact with the slide. You will likely get some leakage of water around the edges and it will be necessary to wipe this excess up with a cloth.
Now that you have your slide prepared, place it onto the mechanical stage and use the metal clips (if present) to fix it in place, with the specimen directly over the hole in the stage. Next, flip on the light at the base of the microscope and look into the eyepiece. To avoid eyestrain on a monocular microscope, try to keep both eyes open although it will take some getting used to. You should see light through the eyepiece and a blurry image.
The next step is to focus the image. Start by adjusting the coarse focus knob (the larger of the two) at the base of the microscope until the image is as clear as possible. Now adjust the fine focus knob until you have achieved an image that is sharp in definition. If you are having difficulties resolving the focus, you can additionally try adjusting the diaphragm to allow less or more light in.
Once you have the microscope focused, you should see your specimen. If you don't see it, try moving the slide around until you are able to locate it. Once you have centered on the specimen, you can increase the magnification by rotating the objective assembly until a larger objective is locked in place below the eyepiece. You will once again have to focus the image, but it should be larger and easier to see the tiny details.
Once you are done viewing your specimen, it is best to clean the slide thoroughly before the water dries and makes it difficult to remove the cover. Congratulations, you have learned to properly use a microscope! Now that you know how to use the microscope, try using more types of specimens.