Viscoelastic flows in porous media are abundant in wide-ranging energy, environmental, industrial, and biological applications. As viscoelastic fluids navigate through a tortuous pore space, the development of elastic stresses under strong flow gradients can give rise to elastic flow instabilities characterized by strong spatiotemporal fluctuations despite a low Reynolds number (Re). Elastic instabilities thus show great potential as tools for improving the transport of mass and solutes in highly confined, spatially complex environments in which flows typically occur at negligible Re. However, precisely how fluid rheology and pore geometry influence the onset and features of elastic instabilities remains unclear—primarily due to challenges with quantifying in situ flow behavior—which limits their practical implementation. In this dissertation, we address these gaps using milli-/microfluidic platforms that enable simultaneous experimental access to pore-scale and macroscopic flow behavior. We first develop a dimensionless criterion to capture the onset of multistable flow states for polymer solution flow in one-dimensional (1D) pore arrays, enabling prediction of flow behavior for fluids of varying degrees of shear-thinning and elasticity. Next, we examine polymer solution flow in precisely-fabricated, 3D consolidated ordered sphere packings, revealing that stagnation points generated at consolidations between grains locally promote the onset of instability. We then address the anomalous flow-thickening behavior of polymer solution flows in porous media by modifying a power balance resistance model to incorporate the contribution from the transient apparent extensional viscosity, which successfully recapitulates the flow resistance in various ordered porous media geometries. We then demonstrate how flow-thickening, together with local flow fluctuations from the elastic instability, can be utilized to enhance surface cleaning of microplastic-soiled porous media. Finally, we investigate flows of wormlike micellar solutions in a range of confined geometries, where an interplay between elasticity, shear-thinning, and geometry promotes strong flow heterogeneity through the formation of low resistance flow pathways. Altogether, these results highlight the power in leveraging non-Newtonian fluid properties to unlock chaos at low Reynolds numbers and lays the groundwork for pursuing the tantalizing goal of achieving targeted flow behaviors through the inverse design of fluid rheology and porous medium geometry.