Okay, welcome to Austin. I don't know of a mecca, just ordinary quad shops. Woods Fun Center is probably the biggest. It is at the intersection of I35 and Braker Lane, exit 242 or so. They have most quad manufacturers, except maybe Artic Cat. I think they have a couple of the new Can-Am Spyders if you are looking to see one of those. Woods also has a newly opened Harley shop up about 10 miles north of the Fun Center. There is also a Polaris dealer in Georgetown, where you can actually test drive the quads or a RZR. They are beside a Honda dealer, but I still wouldn't call them a mecca. There is a KTM shop down on Lamar, a few miles north of downtown. Not a lot else out there, but Austin isn't noted as a quad type of city.
Focus on live music if you like, perhaps the Continental Club or Antone's. Try some BBQ, Iron Works downtown, or pack a cooler full of beer and head to Salt Lick south in Driftwood. You BYOB, so it is a good atmosphere, and pretty decent BBQ. The bat's should be out, so head down to the Congress bridge over the river around dusk and watch the bats emerge. It's amazing if you get to see them before its too dark. Don't stand under them unless you like bat pee on you. For breakfast, head over to Kerbey Lane Cafe. Get a Paris, TX platter which is migas and french toast. Try South Congress (SOCO) just south of the river for dinner. they have a few good selections, including Guero's for Texmex, or Doc's for more general fair. It is a good neighborhood to just walk up and down, plus they have a cupcake stand that is good for dessert.
Good luck.