The answer to this was, of course, different for different people. There were those who were very firmly in favor of the war, those who were strongly opposed, and those who did not care. In general, though, the attitude towards the war among the colonists was more positive right at the beginning than it would be a little later on in the war.
At the start of wars, attitudes tend to be good and morale high. People expect that their side will win the war easily. In the case of the colonists, they felt that the war would surely not last long because the British would make a deal to end the war.
As the war dragged on and the Continental Army had problems, attitudes soured. But at the beginning of the war, there was a great deal of optimism.