We report on the transport properties of ABC and ABA stacked trilayer graphene using dual, locally gated field effect devices. The high efficiency and large breakdown voltage of the HfO(2) top and bottom gates enable independent tuning of the perpendicular electric field and the Fermi level over an unprecedentedly large range. We observe a resistance change of 6 orders of magnitude in the ABC trilayer, which demonstrates the opening of a band gap. Our data suggest that the gap saturates at a large displacement field of D ~ 3 V/nm, in agreement with self-consistent Hartree calculations. In contrast, the ABA trilayer remains metallic even under a large perpendicular electric field. Despite the absence of a band gap, the band structure of the ABA trilayer continues to evolve with increasing D. We observe signatures of two-band conduction at large D fields. Our self-consistent Hartree calculation reproduces many aspects of the experimental data but also points to the need for more sophisticated theory.