Hi Andrew,
Test cases can only be directly associated to one workproduct at a time. To assocaite an existing test case to a story, you would do this by editing the Test case, and click the work product link to select the item in which you want associated. To have multiple test cases associated to a story, you would need to create the test cases from the Test case link in the left sidebar of the story detail page. This would associate all the test cases to the story. In the event you want the same test associated to more than one work product, you would need to copy the test case and associate the copy.
Test Sets are designed to be used for regression testing - where you would repeat the tests in more than one iteration. In this way, you can group the tests together, then the Test set can be scheduled into mulitple iterations.