I recommend you read "Schrödinger's Kittens" by John Gribbin. On page 1 he discusses the most famous experiment that demonstrates the wave properties of light - Young's double slit experiment. Imagine two narrow slits in a board, close together. Shine a light on them and you get a pattern of dark and light lines on the screen beyond that can only be explained by assuming light is a wave. The waves coming from both slits interfere (wave peaks sometime coincide, sometimes clash with troughs, leaving the light-dark-light-dark pattern).
http://physics.about.com/od/lightoptics/a/doubleslit.htm
Faraday thought the universe was filled with a material medium (like the ocean!) that carried the lines of force in the electromagnetic field (he invented those concepts as well). He called this "ocean" the plenum. Think of an olympic* rower; he pushes on the water sending a force through the water and rocketing backwards himself. In that way, Faraday thought of force progressing through the plenum. He thought of the plenum as consisting of microscopic objects, like water molecules, which were the fundamental mechanism of force progression (by action/reaction -- Newton' s third law). He considered light to be due to regular (wave-like) motions in the line-ordered force-carrying particles of the plenum. Phew! Take a pretty picture break:
http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html
Now it gets complicated. There is no plenum. Therefore you cannot visualise directly the reality of what is going on. You can imagine water waves easily because you have experienced the material medium of water and the movements in it.
Notice the above responses to your questions mostly point to the visual effects of EM waves (TV picture, diffraction patterns on the screen...) Visualising these is no problem (you can see them!) But how can you visualise the waves directly? How can they be there when there is nothing material to support the waves, you ask? Gribbin suggest the only way is to visualise what is going on with models and analogies (realising that they are, and can only be, models and analogies) and trust in Maxwell's equations to give you the best account of EM waves because they predict the results of experiments (e.g., the exact position of wave peaks in diffraction patterns). Gribbin (p.66) makes this clear.
He suggests you visualise an EM wave through waves in a stretched rope. Remember that a changing electric field generates a magnetic field, and vice versa. Shake the rope so you get vertical ripples, think of that as the electric field. Because it changes it generates a magnetic field, think of that as at right angle to the changing electric field. That, in turn, generates the electric field. The two changing fields march hand in hand, down the "rope". To improve (?) Gribbin's analogy, take away the rope, but keep the ripples. Voila! Electromagnetic waves propagating through the non-material medium of empty space. David Blaine eat your heart out.
Note, as well as trying to visualise waves in a non-existing plenum you also need to visualise fields of force in empty space:
http://en.wikipedia.org/wiki/Field_(physics )
Not an easy task, visualising this stuff. It occupied the best minds in physics between, and including, Faraday and Einstein, before an acceptable picture was cobbled together. Conveying that picture is not easy, and physics teachers need to try harder in conveying it to the uninitiated (as the drop out rate in physics shows!)
P.S. The double-slit experiment also provides the neatest entry into the even greater mysteries of the quantum world. But that's another story -- with good tellings by Gribbin for the layman, and Feynman in his lectures.
* Go Britain!
