The surface quasi-geostrophic (SQG) equations are a model for low-Rossby number geophysical flows in which the overall dynamics are governed by buoyancy evolution on the boundary. The model can be used to explore the transition from two-dimensional to three-dimensional mesoscale geophysical flows. We examine SQG vortices and the resulting flow to first order in Rossby number, $O(Ro)$. This requires solving an extension to the usual QG equation to compute the velocity corrections, and we demonstrate this mathematical procedure. As we show, it is simple to obtain the vertical velocity, but difficult to find the $O(Ro)$ horizontal corrections. Chaotic transport due to three SQG point vortices is studied with Poincar {e} sections and the Finite Time Braiding Exponent (FTBE). This chaotic transport is representative of the mixing in the flow, and these terms are used interchangeably in this work. Changes in transport from $O(Ro)$ vertical velocity terms are also examined, though without $O(Ro)$ horizontal velocities this is not a true solution to the governing equations. We then consider the SQG elliptic vortex solution developed by Dritschel (2011), in which all $O(Ro)$ velocities can be calculated. Results show that SQG point vortices exhibit greater mixing at the surface than classical point vortices. There appears to be a minimum FTBE near flow regime boundaries, and generally mixing is greater when the energy of the system is greater. There is also a critical depth below which the FTBE decreases sharply. Finally, including vertical velocity in the point vortex solution increases the observed mixing.