I've enjoyed St. Edwards Park for a nice day hike. Close to Austin in Spicewood springs. Fairly easy and scenic.
https://austinparks.org/stedwards-park/
Mt. Bonnell is the highest place in Austin and like to go there for the views of the Lake and city. It's not that much of a hike really, but a place to go that's pretty special and enjoy striking views. Drive along the road to the South and look at the large, beautiful houses along the Lake on the south side as you move along towards downtown Austin. Some great views on the winding and up-and-down road.
Google the topic and several more options for hikes will come up mostly to the West of Austin.
There are also some pretty beautiful places to hike within the Austin City limits like along Shoal Creek Blvd by 38th street.
Walking along the river by downtown is excellent. You can easily walk miles by Zilker park, back towards Lamar, to the Stevie Ray Vaughn statue and even go further to the East side of I-35.
A'Man