The City Museum (Av. Central Poniente and 2 Calle Pte. Nte., no phone, 9:30 a.m.–8 p.m. Mon.–Fri., until 1 p.m. Sat., free) has a handful of rooms exhibiting mostly antiques and domestic memorabilia from Tuxtla [1]’s early days. Signage is in Spanish only, and not terribly engaging at that, but some of the old photos and curios—most loaned or donated to the museum by longtime local families—lend a certain humanity to times past.
There are also documents and artifacts from periods of upheaval, including the Mexican Revolution (in which the Hijos de Tuxtla, or Sons of Tuxtla, played a small but notable role) and the 16th- and 17th-century tug-of-war between colonial powers in present-day Mexico and Central America for control over Chiapas [2]. Upstairs is a large but unremodeled gallery used for temporary exhibits.
Links:
[1] http://www.moon.com/destinations/chiapas/tuxtla-gutierrez
[2] http://www.moon.com/destinations/chiapas