Solution 1:

Small children have a particular writing style that teachers often mark as wrong.

We had a field trip. And we went to the zoo. And we saw monkeys. And they were funny. And then we went home. And the bus was noisy.

Nobody thinks that's a well-written story. So the teacher circles all the "And"s and says "don't start a sentence with and". But somehow we all internalize that as a rule for all of life — which it isn't.

Solution 2:

It is perfectly all right to begin a sentence with a conjunction. It is a special form of emphasis, used to elevate a clause to a position of more influence and importance.

I hold that all beets are red. And I will stick to that belief until you show me a green beet.

We were tired, hungry, and exhausted. But we were home.

It can also be used as a summation of previous statements.

[Blah blah blah ... fairy tale or fable ... blah blah] And that is how the elephant's nose grew into the long trunk it has today.

Solution 3:

I would offer an expansion on those answers concluding that it is not forbidden to start a sentence with "And." The examples thus far are all short sentences which are arguably suited to merging into a single sentence per Chris Browne, excepting the strong emphasis example offered by Sunshine.

My own frequent usage of "And" is associated with two contexts. The first is long sentences that do not easily bear further continuation, yet which are incomplete in their intended task. They require a further thought that, though necessarily connected, is sufficiently different to sensibly permit a new sentence. The "And" makes the connection of the two thoughts less burdensome than would be the case if all were combined into one sentence, however grammatically correct.

The second context is in speeches I write for politicians and business people. While there is obviously a distinction between what is fit for spoken versus written English, the gap need not be large between formal speeches and the written form. And so I say speeches should be written with a mind to being read more frequently than ever they will be spoken.

Now the "And" that starts the previous sentence can be seen as completely superfluous. A sharper writer would omit the entire entrance "And so I say," to achieve that vaunted goal of using the fewest words possible. But I do not acknowledge a linguistic famine that mandates such strict rationing.

I pose no defense against a charge of bad style. To me it is pleasing. And to my clients it has been effective. Or so it would seem by the fact that they return to the source of this bad style.

I can write compactly. I do, as required.

But, when the opportunity presents itself, I luxuriate in languid, listless, indulgent sentences, that succeed in communicating meaning, not for the miserly sake of communicating meaning, but to the more generous end of letting the language dance. And that can take the form of using conjunctions as instruments of pacing, or of bridges over which the waltz can move from one thought to the next.

This is what for me separates language as work from language as recreation.

I can do the work. But I want to play.

Solution 4:

In these cases, "and" and "but" are grammatically the same. Their purpose is to conjoin two clauses or sentences.

"And" as a conjunction usually communicates that the following clause will agree and perhaps expand the previous one.

I like pickles, and I wish everybody liked pickles.

"But" connotes some contrast or unexpected idea.

I like pickles, but I don't love them.

Stylistically, it is not usually considered good practice to begin sentences with a conjunction because a sentence that would begin with "and" or "but" generally connects well enough with the previous sentence that it should connect to it rather than becoming its independent sentence.

Solution 5:

But not ever starting a sentence with a conjunction removes one arrow from your quiver.

I quote (from http://grammar.ccc.commnet.edu/grammar/conjunctions.htm):

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. -- from The New Fowler's Modern English Usage edited by R.W. Burchfield. Clarendon Press: Oxford, England. 1996.