A test strategy is a document that sets up the infrastructure for a given test effort. It will include things like the required test environment, how defects will be managed, what the acceptance criteria is, etc. It is a high-level document that explains to a project team (and management) how the testing effort will be conducted and managed.
Test design can mean a variety of things, but is is often the actual structure of your test requirements, cases, or scripts. In other words, it's the infrastructure for your test assets. It may be defined within one section of your test strategy.
The purpose of a test strategy is to budget for a future test session, including identifying if any specific facilities must be arranged to make the testing feasible. This should also figure high in the risk management process for a project, to see if it makes sense to invest in different testing if the cost to do so exceeds the cost of failure, for example.
The purpose of the test plan is similar to system design specifications: to outline how the testing should be coded in terms of test cases, scripts, etc., what scenarios and conditions must be used, and what a reasonably optimal schedule looks like.