How Fast Do Radio Waves Travel Through The Air?

Radio waves travel at the speed of light, same as all electromagnetic waves. The speed is about 300 million metres per second.

I seem to remember being taught in high school physics that radio waves only travelled at 200 million m/s, but can't find anything to back that up.

Yes, radio waves do travel the speed of light. In fact, it's light in an optical fibre that can only travel 2/3 of the speed of light. The photons are travelling the speed of light, but because they're bouncing around in the fibre the end to end transmission speed is significantly slower.

Radio waves and light waves travel at the same speed in a vaccum but in air they travel slightly slower due to the refrative index of air.  Refractive index of air varies with height, air pressure, water vapour and frequency but is about 1.0003 so the correct answer is  about  2.999 X10^8 ms ( ie 3 x 10^8 for most practical purposes).  Fibre optic cable is not just slower due to the boucing of light in the cable some cable (single mode) the light passes straight through.  The refrative index of glass is much higher than that of air so the light travels slower in glass.

2.9979x10^8 m/s

They travel at the speed of light.

Approximately 299,792,458 meters per second.

".... Does that mean that electric and magnetic field occupies the whole universe?"

I think that inside a Faraday cage the answer is "No".

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