Layered transition-metal trichalcogenides with the chemical formula $AB{X}_{3}$ have attracted recent interest as potential candidates for two-dimensional magnets. Using first-principles calculations within density functional theory, we investigate the magnetic ground states of monolayers of Mn- and Cr-based semiconducting trichalcogenides. We show that the second and third nearest-neighbor exchange interactions (${J}_{2}$ and ${J}_{3}$) between magnetic ions, which have been largely overlooked in previous theoretical studies, are crucial in determining the magnetic ground state. Specifically, we find that monolayer ${\mathrm{CrSiTe}}_{3}$ is an antiferromagnet with a zigzag spin texture due to significant contribution from ${J}_{3}$, whereas ${\mathrm{CrGeTe}}_{3}$ is a ferromagnet with a Curie temperature of 106 K. Monolayers of Mn compounds (${\mathrm{MnPS}}_{3}$ and ${\mathrm{MnPSe}}_{3}$) always show antiferromagnetic N\'eel order. We identify the physical origin of various exchange interactions, and demonstrate that strain can be an effective knob for tuning the magnetic properties. Possible magnetic ordering in the bulk is also discussed. Our study suggests that $AB{X}_{3}$ can be a promising platform to explore two-dimensional magnetic phenomena.