I just want to know if elementary particles, such as a quark, can entangle with another elementary particle of the same kind. If so, does entanglement for elementary particles follow the same guidelines that entanglement between two normal particles (such as an electron)? In other words, is elementary entanglement different from normal particle entanglement?
Both quarks and electrons are part of the Standard Model, our currently accepted theory, and both of these are elementary particles.
Entanglement (QM) is a physical phenomenon, that occurs when pairs or groups of particles are generated, interact or share spatial proximity in a way that the quantum state of each particle cannot be described independently from the state of the others.
Entanglement has been experimentally proven between:
elementary particles like photons, neutrinos, electrons
even small diamonds
The answer to your question, yes it is possible to entangle two different types of particles, and entanglement between elementary particles is the same quantum entanglement that you see between non-elementary particles.
Quantum mechanics is a particle theory, in which objects are described as "particles" instead of "fields" as in QFT. The objects studied could be molecules or atoms, even subatomic particles, depending on the scale you choose. This is same as describing the earth as a point of mass when studying Kepler motions. Here's a link about describing quarks in QM https://public.lanl.gov/mparis/qmp.pdf