This work studies the Van Hove singularity induced novel electronic phenomena in rhombohedral trilayer graphene, including magnetism and superconductivity. Rhombohedral trilayer graphene is a two dimensional crystal where three layers of carbon honeycomb lattice are stacked to form a rhombohedral lattice in its three dimensional extension. The energy band structure of rhombohedral trilayer graphene features van Hove singularities, saddle points in the energy band structure where the density of states diverges. A perpendicular electrical field can be applied to induce interlayer potential and to open up an energy gap at the charge neutrality point. The gap opening modulates the density of states profile, making the divergence at the van Hove singularity robust against fluctuations from finite temperature and disorder effects. Like many other systems, the diverged density of states induces Fermi surface instability. On the one hand, it drives spontaneous ferromagnetic polarization of the electron system into one or more spin- and valley flavors. On the other hand, it allows the bounded states where electrons couple into pairs to have significant lower energy than the normal ground states, leading to superconductivity observable at the experimentally reached temperature. In this work, we fabricated dual graphite-gated rhombohedral trilayer graphene van der Waals heterostructures that allow cryogenic electrical transport measurement and penetration field capacitance measurements and systematically studied the gate voltage modulated magnetic phase transitions and superconductivity. A rigid band model is built that well captures the spin- and valley- phases. For superconductivity, we provided several possible explanations. In addition to gate voltages, we also induced a moiré potential to the system by aligning the lattice of the rhombohedral trilayer graphene with the dielectric material hexagonal boron nitride. We found that while the moiré potential only affect the electronic phase diagram on a perturbative level, several novel phenomena including the breaking of lattice symmetry is induced.