If you have a credit card or debit card from Bank of America, this is your lucky weekend. Every month BOA provides free admission to over 175 museums nation-wide to their customers. In Austin, the affected museums are the Bullock Texas State History Museum, the Thinkery (at Mueller), and Contemporary Austin (at Laguna Gloria and downtown).
Going farther afield, if you head down to San Antonio you can enter the Witte Museum, to DoSeum (museum for kids), the McNay Art Museum and the San Antonio Museum of Art.
Houston and Ft. Worth also have museums on the list.
Here is the fine print:
Offer valid the first full weekend (Sat and Sun) of the month. Photo ID and any valid Bank of America® or Merrill Lynch® credit or debit card must be presented. Free general admission is limited to the cardholder at participating institution. Museum hours and participation varies. Excludes special exhibitions, ticketed shows and fundraising events. Not to be combined with other offers.