How did magnets first come about?

Natural magnets, called "lodestones", were found in iron ores (magnetite) from the ancient region of Magnesia, hence the name "Magnet".

This I knew before reading your question. Wikipedia anwers a question that I had never really considered:

"The other question is how lodestones get magnetized. The Earth's magnetic field at 0.5 gauss is too weak to magnetize a lodestone by itself. The leading theory is that lodestones are magnetized by the strong magnetic fields surrounding lightning bolts. This is supported by the observation that they are mostly found near the surface of the Earth, rather than buried at great depth."

See here. Magnetite $(textrm{Fe}_3 textrm{O}_4)$ is ferromagnetic and can be magnetized. Naturally occurring, already magnetized, versions of this material are called lodestone.

How lodestones itself gets magnetized is still somewhat of a mystery as far as I can tell, though the question is addressed here

The name of the electron indicates that these subatomic particles have an electric field around them.

Electrons also have a permanent magnetic field around them and this fact is obscured by the fact that we have mentally assigned only the electric charge to the electron, but not the magnetic dipole. We measured, calculated and fixed the electrons magnetic moment (NIST reference). It is a constant value.

However, we are not prepared to conclude that a macroscopic magnetic field in permanent magnets is nothing more than the sum of the aligned magnetic dipoles of the electrons involved. We only explain the occurrence of magnetic fields by electric currents. Which is fine for inductive processes, but unnecessary for permanent magnets.

As you know, permanent magnets lose their magnetisation above the Currie points. In return, materials acquire a macroscopic magnetic field at cryogenic temperatures.
This can be explained by the dependence of the chaotic motion and oscillations of subatomic particles on temperature. At certain temperatures, a self-alignment of the magnetic dipoles of the electrons occurs. This process can be favoured by the transition from a liquid to a solid phase. Or, in the case of Bose-Einstein condensates, from a gaseous phase to the BEC state.

the first magnet couldn't be made with that?

You are able to "build" a magnet yourself. Take a material with unpaired electrons (with 1, 3, 5, ... electrons in the outer orbital) It works best with some rare earths. If you make them hot, they lose their magnetic properties. If you cool them down, the electrons involved align themselves through their magnetic dipoles.

In practice, such compounds (materials made of different elements) are ground into a fine powder and then pressed, at the same time an external magnetic field is applied. This aligns more electrons and the artificial magnet becomes stronger. But be careful, such magnets are under strong tension. Do not expose them to shocks, otherwise they will break explosively.