Landmarks
Shaolin Temple
Famous for its connection to Shaolin Kung Fu.
Erqi (February 7) Memorial Tower
Museum inside a rather garish double pagoda that looms over February 7 Square. Inside there are exhibits explaining the city's development.
Shang Dynasty Wall
You can see remants of the wall in the east side of town. Be warned: these mounds are not terribly impressive.
Museums
Henan Provincial Museum
This monolithic pyramid-shaped museum is worth a visit. Learn about ancient civilizations in the area most of which were not based in Zhengzhou, but nearby cities such as Kaifeng and Luoyang, and even see a dinosaur bone or two in the area. The museum is one of the three key museums in China along with one in Beijing. If you are into modern art, this is not the place for you. It generally has acceptable squat toilets on the first floor. There is an English audio tour you can rent, and there are cheapish lockers to store your stuff. Come to see old pottery from the Song dynasty, old bells, bronzes, things like that, but anything more recent does not have English captions. Across the street is a coffee house that makes ok tea and coffee for about ¥30 and has comfortable chairs.
Henan Museum of Science and Technology
Fascinating insight into Chinese children's education. Play on crumbling science exhibits, gawk at the garish space tributes and meet local families.
Parks and nature
Yellow River Scenic Area
Scenic area consisting of five scenic spots, Five Dragons Peak, Camel Mountain Range, Yueshan Temple, Stone Figures, and the Ancient City of Liubang and Xiangyu.
Renmin Park
Has a theme park inside with a quaint Ferris Wheel that is worth riding.
Century Park
Another park worth going to. It is quite a huge amusement park. Be aware most rides have a 90kg limit for anyone using them, sometimes staff can be strict and refuse to let you ride even after you have paid to get in.