Do we know what a radio wave looks like?
Forget the quantum stuff for a moment. If you want to learn about quantum electrodynamics, read QED by Richard Feynman. (You should read it anyway; it may be the only really good pop physics book.)

Classically, an electromagnetic field is a force field that acts on electric charge. It doesn't "look like" something any more than a mechanical push or pull does. One of the things that the EM forces can act on is molecules. They can change the shape of the molecules, or (at high frequencies) even break chemical bonds. That's how you see -- light stimulates a chemical reaction in the cells of your retina, which kicks off a chain of chemical reactions that culminate in brain activity.

When we say that a radio wave can be described as a sine wave, we're talking about how the amplitude of the wave (i.e. the strength of the force) varies over space and time. Sine waves tend to pop up a lot for the reasons Dave mentioned -- they're simple solutions to second-order differential equations, and you can use Fourier analysis to describe other signals in terms of sinusoids. Sine waves are also used to talk about sound, for the same reason.

Most radio waves will not be pure sinusoids, but many are based on sinusoids. For example, the amplitudes of AM radio waves are sinusoids whose amplitude varies slowly. The amplitudes of FM radio waves are sinusoids whose frequencies vary slowly. Here's an illustration, courtesy of Berserkerus on Wikimedia Commons:



Notice that the example signal in this image is also a sine wave. That's not an accident. Sine waves work well as simple test signals. The radiation from power lines would also be pretty close to a pure sine wave.

If you want to visualize a radio wave, imagine being underwater near a beach. The currents aren't visible, but you can still feel moving waves of water as they push you back and forth. That's what radio waves do to the electrons in an antenna.