I like to begin some sentence with "And", especially when I wrote some stories at Reddit. I have someone there who voluntary fix my grammar, and I noticed that he often fix my sentence that begins with "And". But I found beginning sentence with "And" feels more natural to the context of the paragraph. And there, I did it again, I started a sentence with "But". Oh, and another "And" there. Duh. Is there any rule of how to use incomplete sentence correctly? Or is it a big no no ever after in English?
You will often start sentences with and or but in speech to link back to previous things you said. This is OK because in speech you often don't plan out what you say too far in advance and often need to express new thoughts and relate them to previous things you said.
In formal writing, or a formal speech that you are reading, you have (or should have had) time to plan everything out. So this is less acceptable in such a situation.
And and but are meant to link two things. If one of those things is missing and context cannot fill in the blank then you are
I walked to the store. And there I saw Bobby. (First sentence establishes a context, thus from the second sentence we assume Bobby was at the store.)
I walked to the store. And then I got in the car. (Was the car at the store? Typically a car is in a parking lot. So we are missing details here, unless they were in previous sentences.)
I decided to eat dinner and then go home. (The two things here don't depend on anything outside of the context so it's complete.)
The proscription against using conjunctions to start sentences (so called "sentence-initial conjunctions", see google makes you look smart) seems to be taught in school in some places, but descriptive linguists don't agree with it. See Initial coordinators in technical, academic, and formal writing for a nice Language Log post by Mark Lieberman filed under prescriptivist poppycock.