What exactly are EM waves? Wave is just a graph of the intensity of energy at the given point in space right? At a particular point in space, we detect that energy is going up ad then down with each second- when we plot this on a graph against time we get a wave. And 'on the graph only' the wavelength concept is explained right? But with what instrument was this up and down of energy level detected? How did we know that there is an electric field and a magnetic field? Like sound waves travel by creating compressions and rarefactions of the air/gas molecules. How do other forms of energy like electromagnetic radiations actually travel?
At first it does seem unnatural for something to be a field only, no particles actually moving. The velocity of sound is so slow, that you can actually see the propagation of sound visually. Therefore we can visualize it directly. The wavelength of a water wave is directly visible with the eye.
Electromagnetism does not allow that. First, you cannot see electric or magnetic fields. You cannot possible see the propagation of an electromagnetic field from the side. This is because photons do not scatter other photons at a sufficient rate (there is a QED process which can do that, but the cross section is probably minuscule in the visible spectrum).
Once you study quantum field theory, you will see that particles are excitations of fields as well. So as strange as electromagnetism seems, it actually is even stranger than that. But that is the quantum level.
On the classical level, one can think of electric fields to be yet another degree of freedom in space. Just as the gravitational field has a specific value at each point, so does the electromagnetic field.
On the quantum level one can look at the gauge theory. The quantum theory of electromagnetism has a so called gauge symmetry which allows you to multiply the wave function of the electrons by an angle in the complex plane ($\mathrm U(1)$ symmetry). The Dirac equation contains a derivative of the electron wavefunction. A derivative should be thought of a difference quotient here, you substract the value of the field at neighboring points and divide by the distance. The larger the difference across a given distance, the more this field has changed.
Together with the gauge freedom, you will have a problem. Say you rotate differently at point $x$ and $x + \epsilon$ (where $\epsilon$ is the small distance). Then the difference $\psi(x) - \psi(x + \epsilon)$ does not make much sense any more. You are comparing functions which have been “gauged” differently. The solution then is to introduce a parallel transporter to yield a gauge covariant derivative, which removes that problem. This means that instead of $\partial_\mu$ we use $\partial_\mu - \mathrm i e A_\mu$. We must introduce this additional field $A_\mu$ (actually four of them, one for each spacetime dimension) in order to get a unique comparison. This $A_\mu$ will then absorb the change in the gauge freedom.
Now that $A$ is the electromagnetic potential. Taking the exterior derivative of that, one obtains the electromagnetic field-strength tensor. This is a “matrix” which contains the electric and magnetic fields.
To me, this is interpreted like this: The quantum theory of electromagnetism has a certain gauge freedom inherent to it. In order to fully use that freedom, one has to introduce the connection coefficients $A$. Those “take on a life on their own” in the term of electric and magnetic fields.
Also the rotation in the complex plane goes on a circle. The Kaluza-Klein models incorporate electromagnetism by adding another spactime dimension which is wound up to a tiny circle at each point in normal spacetime. By going in loops through this extra dimension, one creates electromagnetic fields.
Another thing is that the potential $A$ follows a massless Klein-Gordon equation. That is a harmonic oscillator (like a spring) without a mass attached. This way one can figure the vacuum being full of little interconnected springs which can be excited just like the surface of water. One has to be careful however. There is no actual ether, no fixed points which can be excited. So the whole thing is still Lorentz invariant, no frame of reference is singled out by this. I cannot picture a relativistic sea of little springs, so I just go with non-relativistic sea of little springs and try not to push the analogy too far.
I hope this helped you with your question. I am sorry that I could not explain it without going through some of the more advanced stuff.
There are a lot of questions intermixed here many of which have already been answered on this site. I'll attempt to give a guide to understand them to some degree.
First of all, you can definitely detect electric and magnetic fields. For electric fields, you just have to take a charged particle/body. If there is an electric field around you, this will then lead to a force on the body. By measuring the force at each point, you can map out the electric field. The same is true for magnetic fields, only this time it is sufficient to take some iron (or any other ferromagnet).
Now this tells you that there are forces on electrically charged objects and on magnets. Upon those measurements, people (such as Ampère, Faraday and Maxwell) developed the concept of an electromagnetic field. It's a theoretical concept and since it happened to describe the results just fine, people started to think about electromagnetic fields.
Maxwell' equations, which combine magnetic and electric fields (a field being just a function that assigns to each point in space some value or vector) admitted a wave-solution. Mathematically, a wave occurs as a solution to the wave equation and it brings with it a definition of wave length and frequency. People like to think of these electromagnetic waves as electric fields inducing magnetic fields and vice versa, but that is only a mental crutch (see for instance this discussion here: Understanding the diagrams of electromagnetic waves). Surely, the wave must carry energy - you can theoretically predict the energy content of the wave (see e.g. here: The energy of an electromagnetic wave) and then you can see what experiments (such as Hertz antennas, see the next section) tell you.
How can you map out electromagnetic waves and their properties like wavelength, etc? Well, you can't directly see them, so everything will be done indirectly. First of all, you can do diffraction experiments such as the double slit experiment. From the pattern on the screen you can theoretically predict the wavelength. You can also use antennas to measure the changing electric field. If you have an electromagnetic wave, an antenna will produce electricity (taking the energy directly from the wave), which you can map out in time. If you know the speed of light, you can directly read of the wavelength from there. The fact that an electric current is induced in the antenna also tells us that truly, electromagnetic waves somehow consist of electric and/or magnetic waves just as predicted by the theory.
Lastly, the waves don't travel in a medium, they can travel in vacuum. This has puzzled people for quite a long time and was only somewhat resolved in the 20th century. See also How do electromagnetic waves travel in a vacuum? for more explanations.
Finally, I have to tell you that all of this is only an approximation. You shouldn't think that there is a real electric fields as described in your physics 101 course. The field picture I described, the varyiing magnetic and electric fields, that's all a classical picture developed in the 19th century which can describe nearly all experiments you can do in a school lab. The current and better description of the electromagnetic field is via Quantum Electrodynamics and contains quantum fields.
You can read something about it in this question What is the relation between electromagnetic wave and photon? or this one Is the wave-particle duality a real duality?