Plasma turbulence plays a fundamental role in determining the performances of magnetic confinement fusion devices, such as tokamaks. Advances in computer science, combined with the development of efficient physical models, have significantly improved our understanding of the mechanisms governing fusion plasma turbulence in recent years. In particular, gyrokinetic simulations have demonstrated their ability to capture turbulent transport phenomena observed in the core of fusion devices. However, the application of the gyrokinetic model is challenged by the different plasma properties in the boundary region. The computational cost of gyrokinetic simulations needs also to be reduced to optimize the design of future fusion power plants. This thesis studies the dynamics of turbulent transport in fusion plasmas through the development of a new simulation code, GYACOMO, based on the gyro-moment approach (Frei et al. 2020). This approach is based on expanding the distribution function on a Hermite-Laguerre polynomial basis in the velocity space. The Boltzmann equation is then expressed in terms of fluid-like equations for the expansion coefficients, the gyro-moments. Within a flux tube, we demonstrate that the gyro-moment approach drastically reduces the cost of nonlinear gyrokinetic simulations in comparison to standard finite difference schemes. The number of gyro-moments required for convergence is reduced when the equilibrium gradients and level of collisionality are large. As a first application of the GYACOMO code, we consider the 2D Z-pinch geometry. The role of equilibrium gradients and collisions in the formation of zonal flows is studied. By comparing the Landau, Sugama, Lorentz, and Dougherty gyrokinetic collision models, the gyro-moment simulations reveal the conditions where using a high-fidelity collision model is required. In particular, the Landau and Lorentz models predict a significantly higher transport when zonal flows dominate. Turning to a tokamak configuration, the 3D $s-\alpha$ geometry is then considered. The cyclone base case, a standard benchmark for gyrokinetic codes, is solved with a computational cost about fifty times lower than the state-of-the-art code GENE. The gyro-moment method also resolves the Dimits shift, in contrast to gyro-fluid models. The possibility of using simplified collision models is shown for the experimental parameters of the cyclone base case. This thesis also shows that the gyro-moment method bridges the gap between gyrokinetic and fluid models. In particular, considering the hot electron limit, the gyro-moment equation system is analytically reduced to the fluid model of Ivanov et al. (2022). This equivalence is also verified numerically with linear and nonlinear simulations. Finally, the efficiency of the gyro-moment method is leveraged to simulate multi-scale turbulence in the edge of the DIII-D tokamak. Six gyro-moments are sufficient to evolve turbulence driven by the ITG, ETG, and TEM instabilities. Electron-scale turbulence has little impact on transport in the considered parameters. Using a multi-fidelity approach, we demonstrate that increasing triangularity tends to destabilize TEMs, reducing the accuracy of simplified electron models. By comparing adiabatic and hot electrons, the effects of triangularity on turbulent transport are finally disentangled.