The essay format that most high schools teach these days is that of the ubiquitous and oft-overused 5-paragraph essay (generally speaking, an introductory paragraph, three supporting paragraphs, and a conclusion; this is likely the type of essay to which @lookingforward refers). While this format might be effective for writing persuasive essays and book reports, it falls short of the type of narrative structure that works well for an application essay.
The opening sentence of a college app essay should ideally provide some sort of “hook” that will draw the reader in and make him/her/them (I’m still balking at using the politically-but-not-grammatically correct singular pronoun them!) want to keep reading. The opening sentence or two (it needn’t be a full paragraph) should indeed present a theme, image, or idea that ties into what follows and, ideally, should be brought back towards the end of the essay.
As for the rest, contractions are generally fine (if not overused), and some of the suggestions regarding word choice given in the article are spot on.