Welcome to Austin! We have some fantastic quilt shops! My personal favorites are (in no particular order): Honey Bee Quilts (Anderson Mill area), The Cotton Cupboard (Lakeway), and A Quilter's Folly (South Austin). B&B Quilting (Buda) and Poppy Quilt n Sew (Georgetown) are also nice.
I also recommend trying Amy's Ice Cream while you're here
A-Town on Burnet Rd is a fun place to shop for souvenirs and gifts, and there's an Amy's right near it