Zion has a number of petroglyphs left by the Anasazi, but also by the Archaic people, as well as by the Basketmaker, Fremont and Piaute cultures as well.
"Guide to Rock Art of the Utah Region" by Dennis Silfer is an excellent overview of rock art, with detailed descriptions of sites.
South Gate, Petroglyph Canyon, and Cave Valley
A few steps from the South Entrance to Zion is South Gate Petroglyph site, which has interesting designs and animal images -- right under the watchful eye of rangers. Other sites in the park are not well advertised, as park officials have had bad experiences with vandalism. Indeed, Petroglyph Canyon is protected and visitors need special permission from Zion Canyon Visitor's Center to visit the site. Ditto for the Cave Valley pictograph site.
Petroglyphs in Parowan Gap Canyon, North of Zion
Rock art fans should visit Parowan Gap Canyon -- accessible from either Parowan or Cedar City. From Parowan, turn west on to 400 North and travel 10.5 miles to Parowan Gap Canyon.
From Cedar City, take I-15 Exit 62 and follow the signs to get onto U-130. Follow 130 north for 13.7 miles and then turn right onto a paved road near mile marker 19. Follow that road 2.5 miles and you're in the middle of fantastic geometric designs, plus images of humans, mountain sheep, bear claws and more.