Red's in Oak Hill is the nearest range I'm aware of. Hill Country Rifle Range out on Hamilton Pool Road used to host IDPA, USPSA, and cowboy action shooting clubs, but the owner - known locally as Crazy Richard - finally alienated all the clubs that used to shoot out there and ultimately closed down to the public; I understand the range south of Bergstrom Airport actually bought a lot of his shooting benches.
As River Rat mentioned, the nearest gun shop is Sportsman's Finest, on Bee Cave Road just north of 71 , more or less across from the Bee Cave HEB. Their prices aren't the best, but they have a lot of NFA stuff to drool over if you're so inclined, as well as some rather pricey shotguns and rifles and a good-for-the-area selection of pistols.
Red's in Oak Hill usually has a decent selection of new guns as well.
One of the oldest gun shops in Austin is McBride's on Lamar - the sales people are knowledgeable and courteous, but the prices aren't particularly attractive. But they do seem to attract more used guns than many other shops.
Heritage on 183 . . . well, I did quite a bit of business with the original owners, but after one instance of doing business with the new ones I vowed never to darken their door again.
And if you don't mind a little ride, Cabela's in Buda (south on I-35, the better part of an hour's drive from where you're at) has quite the selection - since you mentioned classic S&W revolvers, the last time I looked they even had a couple of "Registered Magnums" in their Gun Library. Bring plenty of $$$ if you want one of these. (Oh, wait a minute - you came from Cali . . . if you sold a house there you're probably flush with cash. Go for it!
) Cabela's prices are OK if you get on their mailing list, join the Cabela's Club, and hold off on your purchase until coupons and promotions come in the mail.Once again, welcome to Texas.