Is it only the spin of a particle that can be entangled with another particles spin? Also is there any good physical interpretation of the spin of a particle? because the rotational invariance of entaglement almost literally blows my mind.
Here is an example of how to entangle something other than a spin with a particle's spin. First, take two electrons and put them in the singlet state so that their spins are entangled. Now take one of the electrons and pass it through a Stern-Gerlach magnet. This changes the particle's position in a way that depends on its spin. But its spin is entangled with the other particle's spin, so now the position of one particle is entangled with the spin of the other.
In general, by letting various degrees of freedom of quantum systems interact in this way, you can entangle pretty much anything you want with pretty much anything else.
Entanglement occurs whenever two quantum systems interact. It is not restricted to spin.
The Wikipedia article goes into some detail about what entanglement means mathematically, but I'd guess from the tone of the question this might be a bit too deep. Basically when two wavefunctions interact the result is a new wavefunction that describes the whole system and that can't be simply split into a sum of contributions from the original two systems.
Re your question about the physical interpretation of spin: this is a duplicate of What are some useful ways to imagine the concept of spin as it relates to subatomic particles?.