The terrain of Austin Woods, in general, is mostly flat. No hiking boots or outdoorsmanship should be required for a leisurely walk in this neighborhood. An older region of the Austin area, the occasional crack in sidewalk is possible, and as mentioned, many portions of Austin Woods have little to no sidewalks.
If interested in any nearby greenways, just outside the reach of Austin Woods, load up your pet in the car and drive a short distance in almost any direction. Steck Valley Greenbelt is just a 4-minute drive away from central Austin Woods neighborhood. Stillhouse Hollow Nature Preserve and Barrow Nature Preserve are just a little further if a 10-minute drive is okay for you and Fido.