Houston's Museum District is one of the best concentrations of cultural institutions in the United States, packing 19 museums into a walkable neighborhood just south of downtown along Main Street and Montrose Boulevard. You could spend several days here and not run out of things to see, and remarkably, many of the museums are free.
The Museum of Fine Arts, Houston (MFAH) is the anchor, with a collection spanning antiquities to contemporary art across multiple buildings connected by an underground tunnel designed by Isamu Noguchi. The Menil Collection is a quieter treasure: a stunning Renzo Piano building housing one of the finest private art collections in the world, free to all visitors. Next door, the Rothko Chapel is a meditative octagonal space hung with fourteen monumental Mark Rothko paintings, open to people of all faiths and none.
The Houston Museum of Natural Science is one of the most visited museums in the country, with excellent dinosaur and gem halls and a popular planetarium. Smaller museums worth seeking out include the Buffalo Soldiers National Museum, the Contemporary Arts Museum Houston (free admission), and the Children's Museum Houston for families. The district is served by the METRORail light rail line, making it easy to reach from downtown. Most museums offer free admission on Thursdays, so plan accordingly if you are visiting on a budget.