When listening in to an AM signals of various frequencies, how do we exactly tune in? Alright, so I read on AM and Double Sideband AM, but I don't the more fundamental idea - the frequencies.

Frequency is just: $ \frac{1}{T} $, where T is period

So difference between 5 Hz and 30 Hz is that period of 1st signal is $\frac{1}{5}$ = 0.2 seconds, and of 2nd signal = $\frac{1}{30} = 0.03 seconds$

and I know in AM you need to demodulate by multiplying by cos(wt) to bring message to lower frequency and then low pass filter to only have message signal etc

but for simplicity sake, let's assume two signals are transmitted, one is at 5Hz the other one is at 30Hz

How do you "listen in"? Because the frequency spectrum, where frequency is x-axis is a bit misleading!



It shows that you can magically shift to the right and just listen in to the needed signal! That's not the case! Frequency is just how fast a signal repeats.
Amplitude is changing according to sine function.

So if you have two signals, one at 5Hz and second one at 30Hz, then they both "are in the air", you can't just magically listen in every $\frac{1}{30Hz} = 0.03 seconds$ and only get signal at 30Hz, you'll also be getting the signal at 5Hz, they'll be overlapping because frequency is just how fast signal is changing.

So how do you listen in? Do you just listen in for everything?

Then you have a signal that's changing every "0.2 seconds (5Hz)" and another one every "0.03 seconds", they'll overlap of course. The only way they don't is if there's only one signal is in the air. I don't see how different frequencies allow to fine tune to one particular signal and ignore others.

Besides, even if there's just one signal in the air, say it's the 5Hz signal, how do you "listen in"? Do you just "listen in" continuously, or a receiver just turns on and shuts off every $\frac{1}{5Hz} = 0.2 sec$????

I watched videos, and they just say "receiver" receives, but I don't get how it receives...
Ok, I think here is a good point to introduce equivalent baseband, because you implicitly are already using it! 

So, what does your cosine 5Hz audio signal look like, if you were to set $f_c=0$, in spectrum?

Exactly, one dirac at +5 Hz, and one at -5 Hz! Hence, when you mix that up to $f_c\ne 0$, you get symmetric sidebands. That works for any real signals (ie. for any audio signal) – and even when you add them. That's where your two sidebands come from – positive and (hermitian) symmetrical negative components of any real-valued signal.

So, in baseband, your sum of 5 and 30 Hz cosines have four spectral components – at -30, -5, +5 and +30 Hz, and mixed up to the carrier frequency you get the same discrete spectral components, but added to $\pm f_c$. (your figure also shows a component at $f_c$, but that's only there if you got a DC component in your baseband signal, and/or you're not suppressing the carrier)

Now, there's very different receiver architectures for AM-modulated audio. None of them "just looks every 1/(audio period)". "Looking every so often" presumes you're building something digital. Most AM receivers (most AM being pretty obsolete by now) are not digital. The simplest detector methods really are just rectifying diodes and a low pass filter to get rid of the RF content – I'll leave googling for "AM diode detector" up to you; there's a plethora of good articles on that out there. This is all continuous-time, so there's no "looking that often" there – it all happens by electronically processing the continuous signal.

Now, assuming we're really aiming for digital here:

That 1/(audio period) wouldn't even make sense for audio – it breaks Nyquist; you need at least twice the audio bandwidth as sampling rate to be able to reconstruct the signal.

What one can build is simply a mixer with $f_c$, which will first (continuous-time!) multiply with a harmonic of that frequency (effectively, a complex sinusoid), and then sample (that's the act of looking only every so often) that, and then you get a digital signal that's basically nothing but your original audio sum. 

Note that you've been asking about "how to deal with the fact that you add two sines of different frequencies", but that question is totally independent from the AM aspect of that: you just can. Under the Fourier Transform, i.e. in the spectrum, these are orthogonal, i.e. you can perfectly separate them, both in a computer, in a circuit, or with your ear. That's why you're hopefully able to listen to sounds that aren't made of a single tone. In fact, in reality, single-tone sounds are extremely rare.