We conduct experimental studies on the electrical transport properties of monolayer graphene directly covered by a few layers of CrI3. We do not observe the expected magnetic exchange coupling in the graphene but instead discover proximity effects featuring gate and magnetic field tunability. The tunability of gate voltage is manifested in the alignment of the lowest conduction band of CrI3 and the Fermi level of graphene, which can be controlled by the gate voltage. The coexistence of the normal and atypical quantum Hall effects in our device also corresponds to gate-control modulation doping. The lowest conduction band depends on the magnetic states of the CrI3 and can be altered by the magnetic field, which corresponds to the resistance loops during back-and-forth sweeps of the magnetic field. Our results serve as a reference for exploiting the magnetic proximity effects in graphene.