I have never stayed in Austin proper so I can't suggest an RV park. We usually stay in Fredericksburg, New Braunfels or Blanco and drive into the city. On occasion we have rented a cabin on Onion Creek which is about fifteen minutes south of down town, very nice secluded area but they do not have RV sites.
Midtown RV Park is one I found online that looks nice although its at the base of a freeway interchange so noise might be a concern.
Pecan Grove RV Park also came up and they are on the southside of town but I have never seen the park and they have no website. Google earth shows them covered in trees so its hard to see the park.
If you are not stuck on staying in Austin, I can list some nice parks that are close to the area (30-60 minutes).
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PLACES TO EAT
Hulahut Tex-Mex meets Hawaiian. Very good food and adult drinks. Itโs on Lake Austin, so the scenery is nice.
Texas Chili Parlor An institution for the Capitol crowd. You can walk to the Capitol after lunch here. Definitely not a chain. Burgers, chili, etc.
TEX MEX
Tex-Mex:
Chuy's (where the Bush girl was busted for drinking underage)
El Arroyo (cited in Pat Greenโs song โCarry Onโ; famous for their sign, which changes daily)
Guero'sAustin institution located on South Congress near lots of shops and stores and stuff
BBQ
The Salt Lick is a famous BBQ restaurant in Driftwood which is about forty five minutes west of Austin. Saying this place is HUGE is an understatement. If you go there, call ahead for reservations unless you don't mind an hour or so wait which isn't bad because they have a bar and plenty of seating.
Stubb's Doubleโs as a music venue at night and on weekends.
Iron Works. Decent, not great. Tourists love it for some reason.
BURGERS and PUBS
Hut's Hamburgers. Austin institution. Awesomeness.
The Shady Grove Great stuff.
CAJUN
Shoal Creek Saloon Mmmโฆfried crawfish and oystersโฆ
SOUL FOOD
Hoovers Just awesomeness. Might be too much food, but if youโre going to the Capitol, you can walk it off.
WINGS
Pluckers Try their fried pickles and fried twinkies.
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Austin is known for their wide selection of live music. Pecan Street or its more common name, 6th Street is home to many bars, pubs and restaurant that offer live music.
In late evening you can watch about two million bats fly out from under the Congree Avenue Bridge for their nightly meal. That is some sight to see.
Austin Duck Tour is something that is alot of fun, you can see alot of the city, Lady Bird Lake and learn a little history about Austin.
Depending on how long you plan to stay, there are alot of things to do and see outside of the Austin area but if your visit is short, it would be pointless since there is alot to do in Austin itself.