1. Legislative Reference Library (2nd floor, Rm. 2N.3): The library space is original to the building, and has been in continuous use as a library since 1889. There is much to see, including books, paintings, sculptures, and exhibits that will appeal to all ages, even the young ones. While you're in the library, check out Santa Anna's Chair, and learn about some of the artwork on permanent display. We are open to the public, M-F, 8am-5pm.
2. House and Senate Chambers (2nd floor, east and west wings): Though the Texas Legislature is not in session this year, try not to miss the House and Senate chambers. Both are open to the public, and you'll be able to see where Texas lawmakers debate legislation. In addition, both chambers boast an impressive collection of paintings and photos of early legislatures.
3. Capitol Rotunda (1st floor): Many an Instagram pic has been taken in the Capitol Rotunda, and for good reason. The space offers some of the best views of the Capitol dome interior. In addition, the Rotunda is lined with portraits of former Texas governors, going all the way back to the early days of Texas statehood (spans four floors). 
4. The Capitol Visitors Center: Located in its own building in the southeast corner of the Capitol grounds, the Capitol Visitors Center has ongoing exhibits, and serves as a good jumping off point for your visit at the Capitol. They also offer guided tours.
5. Texas Supreme Courtroom and Texas Appeals Courtroom (3rd floor, north wing): In their early history, the Texas Supreme Court and the Texas Court of Appeals heard cases in the Texas Capitol. Both courts have since moved to new spaces outside of the building, but the rooms are still there, and have been restored to their historic appearance. 

Photo courtesy of the Texas State Preservation Board.