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and. 3 There is a persistent belief that it is improper to begin a sentence with And, but this prohibition has been cheerfully ignored by standard authors from Anglo-Saxon times onwards. An initial And is a useful aid to writers as the narrative continues.  The OED provides examples from the 9c. to the 19c. ...   — Fowler's Modern English Usage (52)

but.  2 Used at the beginning of a sentence.  The widespread public belief that But should not be used at the beginning of a sentence seems to be unshakeable.  Yet it has no foundation.  In certain kinds of compound sentences, but is used to introduce a balancing statement of the nature of an exception, objection, limitation or contrast to what has gone before; sometimes, in its weakest form, merely expressing disconnection, or emphasizing the introduction of a distinct or independent fact.  In such circumstances, but is most commonly placed after a semi-colon, but it can legitimately be placed at the beginning of a sentence and frequently is.  — Fowler's Modern English Usage (121)

and 1. Everybody agrees that it's all right to begin a sentence with and, and nearly everybody admits to having been taught at some  time in the past that the practice was wrong.  Most of us think the prohibition goes back to our early school days.  Bailey 1984 points out that the prohibition is probably meant to correct the tendency of children to string together independent clause or simple declarative sentences with ands...   — Merriam-Webster's Dictionary of English Usage (93)

but 1. Part of the folklore of usage is the belief that there something wrong in beginning a sentence with but: "Many of us were taught that no sentence should begin with but.  If that's what you learned, unlearn it — there is no stronger word at the start.  It announces total contrast with what has gone before, and the reader is primed for the change." — Zinsser 1976  (Several more quotes and examples are included.)   — Merriam-Webster's Dictionary of English Usage ( 211)

Beginning a sentence with a conjunction. There is a widespread belief — one with no historical or grammatical foundation — that it is an error to begin a sentence with a conjunction such as and, but, or so.  In fact, a substantial percentage (often as many as 10 percent) of the sentences in first-rate writing begin with conjunctions.  It has been so for centuries, and even the most conservative grammarians have followed this practice.  Charles Allen Lloyd's 1938 words fairly sum up the situation as it stands even today: "Next to the groundless notion that it is incorrect to end an English sentence with a preposition, perhaps the most wide-spread of the many false beliefs about the use of our language is the equally groundless notion that it is incorrect to begin one with 'but' or 'and'. — The Chicago Manual of Style (5.191)

Beginning a sentence with and or some other coordinating conjunction (but, or or nor) can be an effective means—if not overused— of giving special attention to the thought that follows the conjunction. No comma should follow the conjunction at the start of a new sentence unless a parenthetical element occurs at that point.  (Gregg 1101)