The Old Red Museum is a great place to learn about the history and cultural aspects of Dallas County. There are four main galleries with fabulous films that serve as an introduction to the galleries which then tell you about everything from the dinosaurs that once roamed the county to the latest entertainers born and raised in Dallas. There is a hands-on education center (TEKS related education programs are available for groups for an additional charge), and a temporary gallery that features various Dallas-related exhibits. The building also serves as a beautiful site for special events, and the restored courtroom is a great place to hold meetings.
While the entrance fees are reasonable, I suggest parking at The Sixth Floor Museum in Dealey Plaza, which is only a block away, and save yourself $4. You can put that money towards your visit to The Sixth (& seventh) Floor.