Say that you have a species whose eyes can see anything from radio waves to gamma waves. Let's assume that it's adapted well enough that it can differentiate between these types of waves, and that seeing that much doesn't completely overwhelm it. Besides for the above modification, it's exactly like us in terms of anatomy and physiology. How would society be different with such a species instead of us, if at all?
Let's ignore for a moment how anatomically impossible this is.
Their life would be full of noise. From cosmic radiation to infrared emission every object would glow. There would not be a real night, so society would not form around day or night cycles. They may even be able to see better when the sun is not out spewing noise.
The noise would even be worse as eyelids, even made of lead would not be enough. There would be no darkness. In fact you could see things coming from behind you just as well as in front. They would have to evolve to be able to block as much noise out as possible, but it would not be enough as society evolves.
Technology may also get a huge head start as everyone can see radio waves. We may see primitive radio in the middle ages. Radiation would be discovered by cavemen. They would find rocks glowing in a color they have never seen before, brighter than anything else in the world and spreading in glow to things they touch.
In the end though technology may be stifled. Without a discreet way to send signals life becomes very tough. Who would let a radio station be built, if it was a giant glowing light that you could not stop seeing no matter where you turn? Eventually all society may prize nothing more than to avoid technology and live in cold caves where they can finally get some darkness.
People would stop tanning, cause they know how bad UV rays are for them.
In all seriousness, though, I'd say they'd just appear as far more colors. What makes visible light give color? The fact that we perceive it. If you can perceive the entire spectrum, it's likely they'd all be classified as colors. So you'd have a massive range of colors, some of which would be considered "harmful colors" (uv and shorter). Colors already are used to set a tone, influence a mood, draw attention, give definition, etc. With billions of more shades of colors, they'd be that much more integral to setting tones and moods in society. There would likely be a greater emphasis on color, and visuals would be highly important to everyone (not just guys).
It's a bit tough for me to speculate what colors they would appear to be, but as the visible colors "blend" from red on one side to blue on the other, it's likely radio would be a much more "extreme red", if you will, and gamma rays a more "extreme violet".
People would probably use radio in different ways. When you broadcast a radio program you are using an antenna that emits electromagnetic waves. Such a society wouldn't probably agree to stare at an antenna, the same way you probably don't like to stare at a lighthouse.
X-rays are electromagentic signals too. So, those people would be able to see through some object and through people. So, I think that their medicine would be much more advanced compared to ours, because they would have been able to see through a body since the moment they were able to invent a source of x-rays.
They would have a different kind of art. Think about it. Human artists use all the colors they are able to actually see in pictures. Those people of yours would like to paint in x-rays color, ultraviolet colors, infrared color and so on.
There would probably be many more phrases about colors. Not only "Tonight I'm feeling blue". But instead: "Tonight I'm feeling infrared".
You would be able to know when a person is using his phone, because it would light up, since it sends radio signals. So, at the theaters they would force you to switch off your mobile phone, not just turn down the volume.
The night wouldn't be dark. They would be able to see all the spectrum, so probably we wouldn't have any stories about romantic nights. The only places that are like a "dark room" for your people would be under a mountain.