Charged excitons or trions are essential for optical spectra in low-dimensional doped monolayers (ML) of transitional metal dichalcogenides (TMDC). Using a direct diagonalization of the three-body Hamiltonian, we calculate the low-lying trion states in four types of TMDC MLs as a function of doping and dielectric environment. We show that the fine structure of the trion is the result of the interplay between the spin-valley fine structure of the single-particle bands and the exchange interaction. We demonstrate that by variations of the doping and dielectric environment, the fine structure of the trion energy can be tuned, leading to anticrossing of the bright and dark states, with substantial implications for the optical spectra of the TMDC ML.