Things To Do

A significant part of Austin for us is the Lady Bird Hike and Bike trail which is easily accessed from any location downtown. If you're staying at one of the hotels for the wedding, it will be right across the street. Grab a coffee and take a walk or rent a lime bicycle and cruise the trail.
The other outdoors recommendation is visiting Barton Springs and taking a dip. A great natural spring with a cool view of the skyline. This is required to say you visited Austin.
Tacos:
Marcelino's on the east side for breakfast tacos.
Vera Cruz for migas and fish tacos.
Drink:
Il brutto for a fancy martini
Pinthouse pizza for good hazy IPA
Hotel San Jose for a Michelada
Locations:
East 11th has some fun shopping/eating/drinking (nickel city).
Congress and South congress are located near the capital and the recommended hotels. Lots of shopping, eating, drinking.
Informative:
LBJ Museum
The capital (located near the recommended hotels).
See a show if you can!