Rule #1: Don't use passive voice.
It gives me joy to say that this is not a solid black and white grammatical rule. Yes, my teachers, throughout high school, drilled us that passive voice is the root of all evil, but I'm here to say it ain't. First of all, what is passive voice? Passive voice is when the subject of the sentence is acted upon by the verb (this sentence is in passive voice). An example: My only visit to Budapest will always be remembered by me. In active voice, it reads: I will always remember my only visit to Budapest.
Active voice does sound nicer and is more direct and thus more forcible. Now, before I continue, let me say that if you do not fully understand passive voice, avoid it. Active voice is stronger and better suited for most occasions. The fault in this rule lies in that it is not always better to use active voice. Take this passive voice sentence: The composers of the Rococo period are little remembered today. In active voice: Listeners today hardly remember the composers of the Rococo period. If you were writing about the listeners of music today, stick to the active, but if you were writing about the Rococo composers, what do you care about modern listeners? (Answer: you don't.) Using the passive voice here is better because it focuses our attention on the important subject—composers, not listeners.
Here's an old joke: A man goes to Harvard and approaches a young lady with a question. "Excuse me, miss, but where is the library at?"
"Don't end a sentence with a preposition," she says.
"I'm sorry," he says. "Where's the library at, bitch."
I am only the messenger, not the teller. But the point of the exchange, apart from old and crude humor, is that ending a sentence with a preposition (e.g. for, to, at) is not wrong. That being said, it doesn't sound all that pretty and should often be fixed. Example: Whom did you make that pie for? Technically, that's not wrong, and if you were writing dialogue, don't fix it just to make it sound better—consider your character's grammatical and colloquial intelligence. To remove the dangling preposition, write it as such: For whom did you make that cake? It sounds serious, but both are, technically correct.
Rule #3: Don't start a sentence with a conjunction.
And that was wrong. But your teachers said otherwise. So what? Those sentences all started with conjunctions. And, but, if, or, yet, for are all conjunctions. They join two parts of sentences together. I ate and I drank. It's not wrong to start a sentence with these words, but don't do it all the time. You'll see it a lot in fiction, which does take many grammatical liberties, but the rule is not wrong.
These are probably not the only rules that are not true rules, but they are perhaps the most commonly stated. Just keep in mind that, while left alone they are technically not grammatically incorrect, they often should be removed, especially in a formal setting.