If you're writing a review or critique of the article, then you should definitely be talking about "the article says" or "the author says". But as I understand you, you're saying that you're supposed to write a summary of the article, not a critique. In that case, you should NOT be talking about the article, but about the subject. You should generally not use phrases like "the article says" or "in the first paragraph".
Your summary should follow the general order of the original. Like, if you are summarizing a news story about, say, a crime, and the story starts by introducing the victim, then describing the crime itself, then discussing the police investigation, your summary should also begin by introducing the victim, then describing the crime itself, then discussing the police investigation. Don't re-arrange sections. A summary may drop sections, but you should try to keep all the main points and just drop details. Like in that crime story example, if the original had a long description of who the victim was and where he lived and what he looked like, you might chop that all down to "Fred Jones, a sales clerk from Brunswick".
Of course how much you have to cut things down depends on the length of the original article and the target length of your summary. If you're supposed to write a ten-line summary of a twenty-page article, you'll have to drop a lot more than if the original was one page.
Some people who write summaries try to re-use the original words of the article whenever they can. They'll pick out key sentences and keep those, and pull phrases out of larger sentences. This can help to preserve some of the style of the original, as well as reducing the possibility of inadvertently altering the meaning.
The only time I write things like "the author says" in a summary is if I believe the original writer to be mistaken in his facts or I strongly disagree with opinions that he is expressing or if his statements are controversial or debatable whether I agree with them or not. I don't give a rebuttal -- then it wouldn't be a "summary" any more but a "commentary" -- but I hedge on saying "this is true" and make it "the author claims that this is true". Arguably even that is introducing commentary and no longer a summary. But then, I've seen abstracts on technical articles where the person writing the summary included such hedge words, like, "An experiment was done in which blah blah blah, the author claims to have proven that blah blah" instead of a simple "which proved that".