Improved numerical methods are needed to understand the effect of surfactants in interfacial fluid mechanics, with various applications from thin films and inkjet printing to environmental topics including ocean-atmosphere interactions. In the present study, we provide a three-dimensional coupled Volume of Fluid (VoF) and phase-field-based numerical framework to simulate the effects of insoluble surfactant-laden flows on interfacial fluid mechanics. The framework is validated against analytical cases for surfactant transport and Marangoni stresses. We systematically investigate a single, surfactant-laden rising bubble. The characteristics of a clean bubble rising in a quiescent liquid are governed by non-dimensional numbers, the Galileo number Ga. which compares inertial and viscous effects, and the Bond number Bo, which compares gravitational and surface tension stresses. The effect of insoluble surfactants introduces one more independent parameter, the Marangoni number Ma, comparing the change in surface tension forces due to gradients in surfactant concentration with viscous forces. We apply our numerical methods to investigate the influence of surfactants (through the Marangoni number) on rising bubbles in otherwise quiescent fluids. We observe that an increase in the Marangoni number first decreases the rise velocity before reaching a limiting value at high Ma values. The value of Ma necessary to observe a significant slowdown increases with the bubble Ga number. We discuss the associated surfactant accumulation and the vortical dynamics when a steady state is reached. Finally, we perform full three-dimensional simulations and demonstrate that Marangoni effects can induce a change in the rise trajectory from spiraling to zigzagging for set values of Bo and Ga, in agreement with experimental results.