Graphene is an extraordinary two-dimensional (2D) system with chiral charge carriers and fascinating electronic, mechanical and thermal properties. In multilayer graphene, stacking order provides an important yet rarely explored degree of freedom for tuning its electronic properties. For instance, Bernal-stacked trilayer graphene (B-TLG) is semi-metallic with a tunable band overlap, and rhombohedral-stacked trilayer graphene (r-TLG) is predicted to be semiconducting with a tunable band gap. These multilayer graphenes are also expected to exhibit rich novel phenomena at low charge densities owing to enhanced electronic interactions and competing symmetries. Here we demonstrate the dramatically different transport properties in TLG with different stacking orders, and the unexpected spontaneous gap opening in charge neutral r-TLG. At the Dirac point, B-TLG remains metallic, whereas r-TLG becomes insulating with an intrinsic interaction-driven gap ~6 meV. In magnetic fields, well-developed quantum Hall (QH) plateaux in r-TLG split into three branches at higher fields. Such splitting is a signature of the Lifshitz transition, a topological change in the Fermi surface, that is found only in r-TLG. Our results underscore the rich interaction-induced phenomena in trilayer graphene with different stacking orders, and its potential towards electronic applications.