I am writing an article that is analysing an existing article, that only uses secondary research (research that wasn't conducted by myself - and that I'd obtained from this article). When citing this information in my essay do I cite the same article over and over again when talking about each piece of information? Or cite the article's bibliography? I am only obtaining information from this one article.
Cite the article(s) you use. If you need to talk about that article in terms of earlier ones, cite those.
An example. Suppose the article you are analyzing is "Smith". Then you cite Smith. But if you have to say somewhere that, for example "Smith was led to this conclusion from ... 'Jones'", where "Jones is in Smith's bibliography, then also cite Jones.
Otherwise you are sort of led to citing a chain of publications back to the foundations of the field. Very few papers require that sort of thing and they are very specialized.