Can I start a sentence with 'and' or 'but'?

Your teacher probably told you not to start sentences with 'but' or 'and'. This is generally good advice.

These words belong to the class of coordinating conjunctions (for, and, nor, but, or, yet and so); they are usually used to join two clauses together, for example:

Fox (2004) sets out to explain the rules of English behaviour in Watching the English, but this is not a comparative study, and readers may be disappointed not to find references to their own culture in this popular work.

However, there would be nothing grammatically wrong with separating the above into three sentences even though that would mean starting two of them with 'but' and 'and', respectively:

Fox (2004) sets out to explain the rules of English behaviour in Watching the English. But this is not a comparative study. And readers may be disappointed not to find references to their own culture in this popular work.

It is really a matter of style or register. In academic writing, starting sentences with 'and' or 'but' is considered too informal and should be avoided. Instead, you can use the following:

But > However

And (when meaning 'in addition') > In addition, additionally, furthermore.

And (when meaning 'as a result') > As a result, therefore, consequently.

Don't forget to put in a comma after these terms, for example:

Fox (2004) sets out to explain the rules of English behaviour in Watching the English. However, this is not a comparative study. Consequently, readers may be disappointed not to find references to their own culture in this popular work.