The overriding goal of the British at the start of the Revolutionary War was to maintain a firm hold on the colonies.
Britain, of course, ruled an empire that literally stretched around the world. That empire was forged in blood. It valued its colonies both for geopolitical and commercial reasons. The British understood that to lose the American colonies would not only entail the loss of revenue that came from taxes it imposed on colonials, but it would lose its foothold in the Americas, leaving only its territory in Canada. Given Britain's financial state by the end of the costly French and Indian Wars, the loss of any revenue was not palatable to the Crown; to lose its English-speaking colonies and the revenue they provided was viewed as catastrophic, even with their still sizable empire.
It was that taxation, to help pay for the costs of the French and Indian Wars, that proved so burdensome to the colonies that they revolted.
There would have been a number of different goals that would have been felt by different English people at the start of the Revolution. Their goals would have differed due to their different circumstances and their different political ideals.
For example, a merchant might well have hoped for a very short conflict. This would have been the case because merchants would have wanted trade with the colonies to get back to normal as soon as possible. Such a person would have had no desire for a long and disruptive war.
A Whig might have wanted the conflict to be settled peacefully with a resolution that would have given the colonists more autonomy. Whigs were generally in favor of more democracy. A person who was less inclined towards democracy might have wanted the colonists to be punished severely. Such a person might have worried about increases in democracy. They would have wanted the rebellion suppressed harshly so that no one else would think of doing the same thing.
Thus, there could have been many goals since not all English people had the same political attitudes or economic interests.