Zhengzhou has a growing metro system with multiple lines covering the city centre and major destinations. Taxis are cheap and plentiful; most drivers will not speak English, so having your destination written in Chinese is useful. Didi (China's ride-hailing app) works well and can be navigated with a basic level of app literacy even without Chinese.
For the Shaolin Monastery, the most convenient option is a direct tourist bus from the Zhengzhou long-distance bus station, or a taxi/private car for the 90-minute drive to Dengfeng. Tour agencies at the major hotels offer day trips. The Yellow River Scenic Area is best reached by taxi or arranged transport.