The current study employs a transformation-based analytical technique, namely Riccati modified extended simple equation method (RMESEM) to construct and examine soliton phenomena in a prominent $ (2+1) $-dimensional mathematical model namely Nizhnik-Novikov-Veselov system (NNVS), which has potential applications in exponentially localized structure interactions. The suggested RMESEM uses a variable transformation to turn the desired NNVS into a nonlinear ordinary differential equation (NODE). The resulting NODE is then assumed to have a closed-form solution, converting it into an algebraic system of equations. When the resulting algebraic system is dealt with RMESEM's strategy using Maple, a range of dark and bright soliton solutions in the form of rational, exponential, periodic, hyperbolic and rational-hyperbolic functions are revealed. Some 3D, contour and 2D graphs are plotted for visual representations of these soliton solutions that demonstrate their versatility. The findings deepen our understanding of the NNVS's dynamics, shedding light on its behavior and potential uses.