One of your first stops should be Catalina Island Chamber of Commerce and Visitors Bureau. Here, you'll be able to orient yourself and get some great ideas before you start exploring.
This tourist information center in Avalon comes recommended by travelers, so make sure it features on your to-do list. Don't think that's all there is to do here, though. There are plenty more attractions ready for you to discover after Catalina Island Visitor Center. Catalina Island Golf Course, Descanso Beach, and Catalina Island Museum are not far away.