Greetings for Olli for the best answer.
Of course it is possible to imagine "how the radio waves look like" - or rather - what is the shape of electric (and/or magnetic) field disturbances that propagate in space - despite we can't see them directly. But you need to have a bit of knowledge about them and really rich imagination.
Forget about the quantum and forget about the photons. This is not a level of physics that most can "imagine" in a perceptual way. All those above who mention about photons just don't understand you question or don't know the answer and escape from it crossing the border of something that is beyond people's today's scope. This is as we would talk about the exact shape of atom. What is the shape of a single atom? And what is the shape of a single proton? People have no idea what it is and it is most likely not a little round ball like on school pictures. One can say as long as we don't know the exact shape of the atom we will not understand the correlation between classic electromagnetic wave and elementary particles, i.e. photons, that quantum physics deals with. For today we have to just give up some attempts of embracing something that is un-embraceable, at least regarding ordinary perceptual comprehension of this.
So let's stick to the classical physics and its understanding of a phenomenon which is called an electromagnetic radiation. This is for sure "embraceable", happens in our scale (common radio waves have lengths from 1cm and up) and has been precisely measurable for decades.
However, to surprise, to imagine electromagnetic waves it is very good idea to first 'decipher' and imagine acoustic waves propagation. They are fairly easier to understand. Imagine a single sound wave (a single pulse of it) as a round spherical bubble of highly compressed air in the environment of natural (normal) air and also with the 'normal' air in the center of it. Just a one "layer" of the compressed air arranged in the spherical bubble. This layer does not begin so sharply and does not end sharply. The transition between air pressure values is gentle (like for a wave:). The layer is about 34cm thick (for 1kHz wave) but as I said it faces the surroundings smoothly and ends (on the inner side) also smoothly. Its diameter is let's say 1 meter. And now this bubble is expanding in space in all directions. It's just getting bigger and bigger, but layer thickness does not change - it is 34cm constantly. Just its diameter is growing in all directions around. Its amplitude (the air pressure difference) gradually weakens and eventually it stops existing, disappears. But this was only a single 'layer', a single pulse of an acoustic wave. Now imagine the same bubble growing but after that (exactly 34cm deeper from this one) it appears another one and follows that one by growing spherically, and another one, and another so that we have the whole salvo of them going one after another, moving the serial air pressure disturbances through the space in all directions. This is exactly how the sound wave works and it is worth to remember that they run at extremely high speed which is 343 m/s and really high frequency which is 1000 waves (periods) per a second so that it is faster than somebody fired them from a machinegun regarding both speed and frequency.
Now let's get down to the radio waves. Their shape and propagation have actually the same nature. They are the spherical bubbles (curved layers) that spread in space from their source, one after another. Most important difference from sound waves lie in what the radio waves actually are (what phenomenon do they carry). As we said sound waves carry serial air pressure increments. Their amplitude is the difference between air pressure values in the peaks and in the troughs. That's it. Electromagnetic wave carries electric field increments. One "layer" (or pulse) of it possesses a magnified strength of electric field. Between these pulses electric field value equals zero. So while they travel throughout the space the electric field just alternates between max value and zero. Max - zero - max - zero - max - zero - and so on. Of course all the intermediate values between peaks and zeros are rising and falling alternatively that's why they draw it on pictures as a sine wave.
Furthermore, it is worth to add that electric field is a vector quantity. It means that it has its direction. The direction of the electric field in this case is always perpendicular to the direction of propagation (travelling) of waves. So imagining a single pulse of radio wave as our spherical bubble of the electric field an action of this field is actually directed along our bubble's surface. In other words, the lines of the electric field are curved, parallel to the bubble's curved surface and perpendicular to its radius. Let's consider just a single hypothetical radio wave that travels horizontally. We can assume now that the electric field direction is vertical. And now it comes the thing - electric field direction alternates between pulses. For our horizontal wave - the field in first period goes vertically up and in next one it goes down. So in one bubble it is directed up, in next it is directed down. Still the places between bubbles have field value zero and every bubble has field directed opposite to adjacent bubble's field. We can brief it as: max - zero - min - zero - max - zero - min - zero. An amplitude of the wave is the difference between maximal and minimal (or as we can say - negative) intensity of the electric field. Remembering about all intermediate values we now know why they draw it as a sine wave with the horizontal axle placed in the center (where the field's intensity equals zero). No matter the field direction is up or down - it is still perpendicular to the travel of wave, isn't it? And this is how exactly the electric field is set up in the space between subsequent pulses of waves (or between spatial bubbles which grow one after another).
But there is yet another component which seems to make things really complicated - magnetic field. Actually this is not so difficult to figure out. Magnetic field activity covers the same regions as electric field. They are correlated in phase. In points - or spatial spheres actually - where the electric field is zero - the magnetic field is also zero. In spheres where the electric field intensity has its peaks - the magnetic field intensity has also peaks. In spheres where the electric field has its troughs - the magnetic field has troughs. As you guess that magnetic field is also a vector quantity because its acting lines have direction. The basic difference is that magnetic field direction is perpendicular to both the travelling of wave and the electric field direction. As we imagine our hypothetical horizontal radio wave with the electric peaks vertically up and electric troughs vertically down the direction of magnetic field lines would lie along the line of our sight. The magnetic peaks are then directed towards us and magnetic trough are directed out from us. If we consider a broader area the magnetic field lines must also go along a curve - along a surface of sphere.
I don’t know how much can be understood from what I said:) However the main idea is that these are bubbles of magnified electric and magnetic field which also alternate its direction every second bubble and these bubbles grow very rapidly. As they travel through space by growing the strength of electric and magnetic field weakens (the amplitude decreases), they lose their energy and after some distance covered they finally vanish at all (the same as acoustic waves).
In reality the shape and layout of all these waves (both acoustic and electromagnetic) is much more complicated because of things like reflection, interference, diffraction and refraction. The bubbles reflect from various objects like ground, buildings, trees, cars, walls, furniture and so on. Reflected bubble hits the direct one and affects the shape and exact travelling of each other so the resulting topology of the waves is usually very complex and unpredictable from the perceptual point of view.
To complete the basic physical differences to sound waves which we obviously know is:
- they don't need any medium, they are self-propagating and can travel through both vacuum and many various materials;
- their wavelength can vary a lot but for Wi-Fi it is about 9-15cm so it is quite close to sound wavelength that we discussed;
- their frequency is extremely higher (for example 100 MHz for FM radio or 2.4 GHz for Wi-Fi);
- their speed of travelling is also extremely faster (speed of light);