A long-held tradition of stickler English teachers, the great sin of beginning a sentence with "and" or "but" is, in fact, not a sin at all. To consider why this myth has persisted, we must first look at what "and" and "but" are -- conjunctions. Coordinating conjunctions, to be exact.
A coordinating conjunction joins elements of a sentence that are equally important. (Soanes)
- The cat was black and white.
In this example, "black" and "white" are both adjectives joined by the coordinating conjunction, "and". And, as you might have noticed over time, coordinating conjunctions tend to be placed within a sentence -- the possible root of why so many grammarians and teachers seek to stop their use at the beginning of a sentence. However, there is no strict rule for keeping these coordinating conjunctions detained to the inside of sentences.
While there is no real rule against using "and" or "but" at the start of a sentence, you should still be careful about its overuse. The reason to use these words is for emphasis and dramatic effect -- not mere convenience.
- Bob was having a bad day. And it was about to get worse.
Creating a second sentence, in this example, adds more emphasis to that sentence. Thus, the usage of "and" or "but" at the beginning should only be done as a stylistic consideration and overuse should be avoided, lest you deflate the importance of such breaks.
- Soanes, Catherine "Grammar myths #2: please miss, can I start a sentence with a conjunction?" OxfordWords Blog. 5 Jan. 2012 <http://blog.oxforddictionaries.com/2012/01/can-i-start-a-sentence-with-a-conjunction/>