© 2019 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the »https://creativecommons.org/licenses/by/4.0/» Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. We present an extensive overview of the phase diagram, spin-wave excitations, and finite-temperature transitions of the anisotropic-exchange magnets on an ideal nearest-neighbor triangular lattice. We investigate transitions between five principal classical phases of the corresponding model: ferromagnetic, Néel, its dual, and the two stripe phases. Transitions are identified by the spin-wave instabilities and by the Luttinger-Tisza approach, and we highlight the benefits of the former while outlining the shortcomings of the latter. Some of the transitions are direct and others occur via intermediate phases with more complicated forms of ordering. The spin-wave spectrum in the Néel phase is obtained and is shown to be nonreciprocal, α,k≠α,-k, in the presence of anisotropic bond-dependent interactions. In a portion of the Néel phase, we find spin-wave instabilities to a long-range spiral-like state. This transition boundary is similar to that of the spin-liquid phase of the S=1/2 model discovered in our prior work, suggesting a possible connection between the two. Further, in the stripe phases, quantum fluctuations are mostly negligible, leaving the ordered moment nearly saturated even for the S=1/2 case. However, for a two-dimensional surface of the full 3D parameter space, the spin-wave spectrum in one of the stripe phases exhibits an enigmatic accidental degeneracy manifested by pseudo-Goldstone modes. As a result, despite the nearly classical ground state, the ordering transition temperature in a wide region of the phase diagram is significantly suppressed from the mean-field expectation. We identify this accidental degeneracy as due to an exact correspondence to an extended Kitaev-Heisenberg model with emergent symmetries that naturally lead to the pseudo-Goldstone modes. There are previously studied dualities within the Kitaev-Heisenberg model on the triangular lattice that are exposed here in a wider parameter space. One important implication of this correspondence for the S=1/2 case is the existence of a region of the spin-liquid phase that is dual to the spin-liquid phase discovered recently by us. We complement our studies by the density-matrix renormalization group of the S=1/2 model to confirm some of the duality relations and to verify the existence of the dual spin-liquid phase.