Varies depending upon what region it is made. There are mainly two types - clear soup and miso-based soup.
The clear chicken-broth zōni, which I made, is from the Tokyo area. We prepare chicken broth and add grilled mochi, mitsuba leaves, naruto (fish cake designed in vortex). I made it big (coz, I'm a pig) and I used a different type of fish cake (kamaboko) instead.
The miso-based zōni is from the Osaka-Kyoto area and is made mainly of root veggies like daïkon, carrot, taro. They stew mochi in the soup.