Can you begin a sentence with words like "And" or "But" in just every day writing or is the same rule as formal writing where no sentence can begin with "And" or "But"?
In my opinion, you can begin a sentence with "And" or "But". Whether you should or not may be a different question.
I had expected someone-else to jump on this question, but that doesn't seem to be happening so here goes...
I suspect that some people will claim that a "sentence" that begins with "And" or "But" is not really a sentence. Based on the formal definition of a sentence, they are probably correct. A structure beginning with "And" or "But" is not a complete thought by itself and is therefore not a sentence.
Never-the-less, these structures do get written. What is it that they are trying to convey?
In general, I would say that starting a sentence with "And" or "But" implies that whatever precedes it (usually another sentence) represents a totally completed thought. The "And"/"But" structure is a sudden afterthought, logically related but having suddenly jumped to another train of thought.
Use of "And" or "But" (or "Or") to begin a sentence is jarring to the reader and some writers use it as a stylistic form with exactly this intent. In general over use of "And" and "But" quits being jarring and becomes simply annoying.
If used to follow the POV (point of view) of an individual in a story, it gives the feeling that the individual is scatter-brained. For this reason, you might want to avoid it in your own personal correspondence.