First, we need to separate grammar and style. Grammar and correctness represent the rules of written English that are generally agreed upon and that should be followed when writing. Style represents choice that may be common to a discipline, imposed by an editor (or professor), or left up to the writer.
Grammar errors include comma splices, sentence fragments, and subject-verb disagreement. For the most common errors see, Stanford University Writing Center’s “Top Twenty Errors in Undergraduate Writing.”
Correctness is a broader concept and includes the usage of words and phrases. These can be contested, and they can change over time. For example, if you use “begs the question” to mean “raises the question, then you are using it incorrectly, but this common mis-usage is becoming more accepted (I still reject it). For a list of words to flag, see The Independent’s “58 Most Commonly Misused Words and Phrases.”
Style is more complicated. For example, American historians don’t like the passive voice (“Africans were enslaved”) because it obscures human agency and responsibility (who enslaved them? what was the process?). But passive voice is preferred in other fields, especially when the focus is on the object of the action (“these results were checked against the control”).
More generally, there is nothing ungrammatical about the first person, and academic writers actually use it all the time (even in and sometimes especially in scientific fields where you were taught never to use it). But they use it only when and how they think it is effective–only in certain types or genres of writing. And they may even disagree with each other about those choices. Disciplines change over time. The best general advice I can give you is to ask your professors about particular style choices in their disciplines, and to ask them for the reasoning behind those choices.
[More to come.]