>>16825554 (OP)
Spin is intrinsic angular momentum; you can talk about classically without any quantum mechanics concepts. Angular momentum is a conserved quantity of the dynamics and you simply have to keep track of this additional intrinsic angular momentum along with the orbital kind. If you know what angular momentum is from mechanics, that’s all there is to it. It acts like a little magnetic moment and couples to magnetic fields the way you’d expect any “spinning” charge to.
What makes spin weird and frustrating is how it behaves under a change of coordinates when you start doing quantum mechanics; if you rotate the spin vector of one particle by 360 degrees relative to the other, their wave functions destructively interfere, so the actual state of the system gets rotated only 180 degrees, and you need another full 360 degrees to get constructive interference again.
As a crude analogy, you can imagine the spin state as an arrow tip that rotates attached to the edge of a Möbius strip, so a 360 degree rotation flips its direction while 720 degrees aligns it back with itself. This is what physicists mean when they talk about SU(2) symmetry and double covers of the rotation group (although in full generality spin transforms under the double covers of the Poncaire group)