In spite of what some old-fashioned textbooks say, it is quite acceptable to start a sentence with 'But' in informal written language. In informal and neutral contexts, you can also begin a sentence with 'That said,' or 'Even so' or 'All the same', which all have the same effect. You can also put '.., though' at the end of the sentence instead of 'But' at the beginning.
In more formal contexts, you need to replace 'But' with 'However', followed by a comma. Other formal equivalents are 'Nevertheless' or 'Nonetheless.' You could also consider phrases such as 'In spite of this fact,'..