In the olden days (aka the 1980s), when I was at school, our English teachers told us never to start a sentence with a conjunction. A conjunction is a connecting word such as and or but - or or*.
Conjunctions, we were told, should always come in the middle of a sentence, to link two phrases. For example It was a bright cold day in April, and the clocks were striking thirteen. Or She knew how to give, but not how to receive.
But this a rule that's not a rule. If you don't believe me, read back over the past few sentences and spot the conjunctions at the beginning. You can start a sentence with And. Or But. Or Or, or any conjunction you like. George Orwell - whose opening line to 1984 is above - was no stickler when it came to so-called rules. (He also wrote, in the same novel But it was all right, everything was all right, the struggle was finished.)
So (that's another conjunction) when is it OK to use conjunctions at the beginning, and when is it not? It all comes down to rhythm and impact. Beginning a sentence with a conjunction adds power to what you're saying. Think of the difference in impact between It had been a long, tiring day, and it wasn't over yet. and It had been a long, tiring day. And it wasn't over yet. The first example is gentler, while the second is more powerful. The first implies nothing except that the day is not yet over. The second implies that something is going to happen before the end of the day. Which one you choose depends how much impact you want to have.
If you have a really long sentence, breaking it up into two will make it easier to read. And beginning your sentence with a simple conjunction avoids overly formal language. In an academic essay you have to use phrases like However... or In addition... - otherwise, you can use But or And.
Most of all, consider the rhythm as you write. Read your writing aloud - would it sound better with a long stop in the middle, or a short pause? Write in the same rhythm you'd use to speak. And have fun with language. It's a wonderful gift. Enjoy.
*Conjunctions are small words that link ideas or phrases. Some examples are:
AND BUT OR SO FOR