The progressive and enhanced thermal mechanisms of nanoparticles has motivated researchers to give attention to this topic in recent years. The synthesizing and versatile applications of such materials include cooling and heating controlling processes, solar systems, energy production, nanoelectronics, hybrid-powered motors, cancer treatments, and renewable energy systems. Moreover, the bioconvection of nanofluids allows for some motivating applications in this era of bioengineering and biotechnology, such as biofuels, biosensors, and enzymes. With these interesting motivations and applications, this study elucidated upon the three-dimensional bioconvection flow of a Sisko fluid (base fluid) in the presence of a nanofluid over a stretched surface. The additional thermal features of radiation were also incorporated to modify the analysis. The rheological features of shear thinning and shear thickening that are associated with the Sisko nanofluid were comprehensively studied. The problem was formulated using highly nonlinear and coupled differential equations, which were numerically simulated via a shooting scheme. The salient physical applications of flow parameters were graphically underlined in view of shear-thinning and shear-thickening scenarios. The results showed that a decrease in velocity in the presence of buoyancy ratio forces was more conducive to the shear-thinning phenomenon. The increase in temperature profile due the thermal Biot number and surface heating source parameter seemed to be more inflated in the shear-thinning scenario. A lower motile microorganism profile was noted for the bioconvection Lewis number.