How can I get a list of the topics and queues in a broker?

As of 5.1.0 you can use the new DestinationSource on an ActiveMQConnection to access the available queues or topics or listen to queues/topics being created or deleted.

You can use JMX to browse the available topics and queues in a broker together with the connectors and connections along with their statistics etc.

Or you can access statistics programmatically.

Finally you could just use the Web Console