Flow of viscoelastic fluids in porous media is ubiquitous in biological and industrial processes, and the choice of viscoelastic fluids is of foremost importance for their effect on macroscopic flow and transport properties. In this paper we study the macroscopic properties of flow and transport of viscoelastic fluids through a model porous media by means of Direct Numerical Simulation (DNS). We show that flow of a viscoelastic fluid modeled via a FENE-P constitutive model features three distinct regimes of flow resistance: a plateau at low Deborah numbers, and a shear-thinning phase followed by an abrupt flow thickening above a critical Deborah number, consistent with experimental observations. Congruous to this shear-thickening, we observe the onset of hydrodynamic instabilities resulting in fluctuations in pressure drop and flow resistance. These fluctuations intensify with increasing the fluid elasticity. Finally, we investigate Lagrangian attributes of viscoelastic flow and transport through porous media. We find that although the fluid elasticity broadens the Lagrangian velocity distributions, it does not alter the long-term particle dispersion in disordered porous media.