Where should I place my beans and servlets under Tomcat?

Where you place the beans and servlets depends on which "application" it belongs to. Assuming that you are running your JSP pages under the default "examples" application, you should place your servlets under the path

/tomcat/examples/WEB-INF/classes

/tomcat/examples/WEB-INF/jsp/beans

Both these paths are placed within CLASSPATH by Tomcat's startup script file.