The basic procedure for testing an electronic circuit with an oscilloscope is to attach the ground connector of the scope's test lead to a ground point in the circuit, and then touch the tip of the probe to the point in the circuit that you want to test.
For example, if you want to verify that the output from a pin of an integrated circuit is emitting a square wave, touch the oscilloscope probe to the pin and look at the display on the scope.
Note that you may need to adjust the VOLTS/DIV and TIME/DIV settings on the scope to clearly see the waveform. But once you get those settings adjusted correctly, you should be able to visualize the square wave. If the square wave doesn't appear, you likely have a problem with the circuit.
Never connect the oscilloscope probe directly to an electrical outlet. You're likely to kill your scope or yourself. (If you want to measure voltage from an outlet, just use your regular multimeter.)
The following paragraphs give a few ideas for viewing various kinds of waveforms with an oscilloscope:
To view a simple DC waveform, try connecting the oscilloscope to a 1.5 V battery such as a AA or AAA cell. Set the VOLTS/DIV knob to 2 V, and then touch the probe ground connector to the negative battery terminal and the probe tip to the positive terminal.
The resulting display should be a simple straight line midway between the second and third vertical division above the centerline. (If the battery is dead or weak, this line may be lower.)
If you want to see the 60 Hz sine wave available from an electrical wall outlet, find a plug-in power supply (commonly called a wall wart) that generates low-voltage AC. If you don't have one lying around, you can buy them new at many stores. You can also find them for $1 or so at thrift stores.
Plug the wall wart into an electrical outlet, and then connect the oscilloscope probe to the wall wart's low-voltage plug. Adjust the VOLTS/DIV and TIME/DIV settings until you can see the sine wave.
If you want to see what an audio waveform looks like, find a short 1/8″ audio cable that's male on both ends. Plug one end into the headphone jack of any audio device, such as a radio or an iPod. Then, connect the probe's ground lead to the shaft of the plug on the free end of the audio cable and touch the probe tip to the tip of the audio plug.
After fiddling with the VOLTS/DIV and TIME/DIV settings, you should see a display of the jumbled waveform that's typical of audio signals.
A decent oscilloscope will cost at least a few hundred dollars, and really good ones start at $1,000 and go up from there. So although oscilloscopes are useful, most electronic hobbyists get by without one.
As you build circuits, keep your oscilloscope handy. Don't hesitate at any time to pick up the oscilloscope probe and check out the signals that are being generated at various points within your circuit. Connect the probe's ground clip to any ground point in the circuit, and then touch the probe tip to every loose wire and exposed pin that you can to see what's going on inside the circuit.