This thesis focuses on investigating local and nonlocal transport properties in hexagonal boron nitride (hBN)/graphene superlattice Hall bars and graphene Hall bars proximity coupled to the ferrimagnetic insulator yttrium iron garnet ($Y3Fe5O12$ or YIG). The first part describes in detail the pulse laser deposition of atomically flat YIG thin films onto single crystal gadolinium gallium garnet with a magnetization of 144 emu cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-3</mrow>}$. This part also outlines device fabrication procedures including graphene exfoliation and dry transfer, electron beam lithography and metallization of side-contacts, and finally the electrical setup for measuring local and nonlocal transport in graphene. The second part investigates transport properties in hBN/graphene/hBN superlattice Hall bars with a field-effect mobility of up to 220,000 cm$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ V$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ at 9 K with low charge impurities. By aligning hBN and graphene, a ∼33.7 meV band gap at 9 K is demonstrated at the primary Dirac point in zero magnetic field. Furthermore, the nonlocal resistances approach $h/2e2$, where $h$ is Planck’s constant and e is the electron charge. Nonlocal measurements demonstrate that, below 60 K a spin-degenerate ballistic counter-propagating edge state forms and dominates with a possible secondary contribution from a network of one-dimensional conducting channels with soliton-like domain walls. The spin-degenerate ballistic edge states offer possibilities for electronic applications beyond quantum spin and anomalous Hall effects since a quantized resistance is observed through valley coupling. The third part reports a proximity-induced magnetic exchange field in graphene of the order 60 T by placing graphene on the ferrimagnetic insulator YIG. From electrical transport measurements, the magnetic order and energy gap of the edge modes in graphene are tunable, and a transition between the canted antiferromagnetic and spin-polarized ferromagnetic $\nu =0$ quantum Hall states can be achieved with relatively low magnetic fields ($>6$ T) at 2.7 K. The fourth part summarizes the key results of the thesis.