The Zilker Botanical Garden is gorgeous and worth a visit if y'all want to see some Austin-native plants. Jessica and I are fans, and we hope to be able to keep the various shrubberies we have at home alive long enough to be half as luscious as the ZBG. The Lady Bird Johnson Wildflower Center is also a great visit, although the blue bonnets that Central Texas is known for are only around in the Spring.
Speaking of LBJ, the LBJ Presidential Library is a great place to spend an afternoon. We've been once, and it was lovely to read the letters LBJ and Lady Bird Johnson sent to each during their courtship. Both of 'em have an inspiring legacy to explore, go check it out! Also, Jessica tells me that Sunday, September 22nd is Austin Museum Day (who knew?!)
This is our favorite thing to do in Austin: go brewery hopping around the city where we chitchat about everything and nothing. There are tons of great breweries: Easy Tiger (if you like pretzels), Central Machine Works (for the live music), Lazarus (for a laid-back atmosphere), Celis, Meanwhile, Jester King, and a zillion more. For what it's worth, Andrew's favorite beer is the Live Oak Hefeweizen and Jessica's is Pinthouse's Electric Jellyfish.
It gets friggin hot in Austin during the summer, but luckily at the end of September, the dog days are ova-aaah. Go and take a dip in the cool, spring-fed water, and watch some locals do heroic belly flops from the diving board. Or of course, you could be that hero.