What are the strengths of GET over POST and vice versa when creating an ajax request? How do I know which I should use at any given time? Is it a security-minded decision?>
GETs should be used for idempotent operations, that is operations that can be safely repeated more than once without changing anything. Browsers will cache GET requests (for normal and AJAX requests)
POSTs should be generally be used for non-idenpotent operations, like saving something. Although you can use them for other operations if you want.
Data for GETs is sent over the URL query string. Data for POSTs is sent separately. Some browsers have a maximum URL length (I think Internet Explorer is 2048 characters), and if the query string becomes too long you'll get an error.