In the past, English teachers used to preach that one should never start a sentence with conjunctions like and or but. Does this rule still apply today?
Not entirely. It is already acceptable to start sentences with such conjunctions. Some authorities, in fact, even defend that for some cases conjunctions will do a better job than more formal constructions. Here is a quotation from
Ernest Gowers addressing the usage of and on the beginning of sentences:
That it is a solecism to begin a sentence with and is a faintly lingering superstition. The OED gives examples ranging from the 10th to the 19th c.; the Bible is full of them.