I have so far derived the link between special relativity in a straight wire with a current and the Biot-Savart law. Here's my problem. All this is based on the fact that the magnetic field for some mysterious reason should go around the wire, instead of straight down/up from/to the wire like the electric field. I understand why the electric field goes like it does, but I don't understand what the magnetic field is. I only know the formulas that we have been taught in school, but they only explain the connection between the electric field and the magnetic, not why the magnetic is like it is. And then there is this magnetic flux term which got described as the magnetic flux density moving through a given area. I think the problem lay in the fact that I fundamentally only understand the effects of magnetism not what it is. (Btw I'm in high school if it helps anyone to know at what level they should be explaining at. I really hope someone can help me understand this)
I fundamentally only understand the effects of magnetism not what it is.
Uhmm. Do you understand "what is" electric field? As a general note, it's not advisable to put such questions in physics. Physics isn't about what things are but rather about how they behave (in a general and variable meaning).
the magnetic field for some mysterious reason should go around the wire
Well, this is an experimental fact. Direction of magnetic field is said by a magnetic needle.