Oh, my home! Unfortunately I was not an active cyclist when I lived there but most of my friends were so: if you can ride the Greenbelt, do it. If you can ride to Barton Springs and go swimming, do it. If you can do a longer ride out into Hill Country, do it. Please avoid the major road ways if there isn't a bike lane (though there was supposed to be a lot of fresh paint put down in the past several months)...
And if you hit it, you might as well do the Austin Critical Mass... see if you notice it being significantly different from Chicago's.
http://critical-mass.info/austin.html