If I wanted to find out what kind of electomagnetic waves "travel" through my room at which frequency, what kind of equipment would I need? Suppose I want to view frequencies from 0 Hz to 6 GHz.
The oscillating magnetic field associated with an EM wave will induce a voltage in any electrical conductor that it passes through. So in principle all you need to do is stretch a piece of wire across your room then measure the voltage across it. However, as usual, the devil is in the detail.
If you've ever listened to a radio in your room then you've done this experiment for a limited range of wavelengths. Radio is transmitted as an electromagnetic wave, the wave induces an oscillating voltage in the aerial and the electronics in the radio amplify and process this voltage to extract the sound. But a radio will only receive a limited range of wavelengths, and this is the main problem you'll run into. If you want to receive and measure EM waves of all frequencies from zero to 6GHz you'll need some specialist kit.
The usual way this is done is with a spectrum analyser, but you'll pay a lot for a good quality one. You could do it on the cheap by looking for a second hand oscilloscope and building your own filters to select specific frequencies for measurement.
Whichever way you do it, you'll still run into other problems. It's very difficult to measure low frequency waves because the voltages they induce in your aerial will be very low. Ultra low frequencies require very large aerials to detect them.
(You can also monitor the web pages your neighbor is surfing on. Similar tricks work with what you type into a keyboard, if you know the keyboard.)