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?