I think your understanding is fundamentally correct.
A test plan is just that -- a test plan. It is how you plan to go about testing an application or change to an application as a whole. The plan might include a description of what scenarios you need to cover, whether they will be automated tests or manual tests, whether you will do performance testing, in what environments (os, systems, browsers, etc) the tests will cover, and so on.
Wikipedia says this about a test plan:
A test plan is a document detailing a systematic approach to testing a system such as a machine or software. The plan typically contains a detailed understanding of the eventual workflow.
A test suite is a collection of test cases. Think of a test case as a file on disk, and the suite is a folder (though they might not be actual folders and files). A test suite is a way to organize all of your individual test cases.
Wikipedia says this about a test suite:
...a test suite, less commonly known as a validation suite, is a collection of test cases that are intended to be used to test a software program to show that it has some specified set of behaviours.
A test case is what is actually performed during testing. A test case could be a single automated test, or it could be instructions for performing a manual test.
Wikipedia says this about a test case:
A test case, in software engineering, is a set of conditions under which a tester will determine whether an application, software system or one of its features is working as it was originally established for it to do