We investigate the proximity spin-orbit and exchange couplings in ABA and ABC trilayer graphene encapsulated within monolayers of semiconducting transition-metal dichalcogenides and the ferromagnetic semiconductor Cr2Ge2Te6. Employing first-principles calculations, we obtain the electronic structures of the multilayer stacks and extract the relevant proximity-induced orbital and spin interaction parameters by fitting the low-energy bands to model Hamiltonians. We also demonstrate the tunability of the proximity effects by a transverse electric field. Using the model Hamiltonians, we also study mixed spin-orbit/exchange coupling encapsulation, which allows us to tailor the spin interactions very efficiently by the applied field. We also summarize the spin-orbit physics of bare ABA, ABC, and ABB trilayers, and we provide-along with the first-principles results of the electronic band structures, density of states, spin splittings, and electric-field tunabilities of the bands-a qualitative understanding of the observed behavior and realistic model parameters as a resource for model simulations of transport and correlation physics in trilayer graphene.