How did Native Americans react towards the fighting between Britain and France in North America during the years 1756-1763?
The period you specify is the French and Indian War, also known as the Seven Years War. Native Americans fought on either side of the war, not so much because they agreed with or liked the Europeans with whom they fought; but because they did not like and were at war with the Native Americans on the other side. A little known fact is that the war in America actually began as a conflict between two Indian tribes.
Samuel de Champlain, the French explorer, had intervened in a dispute between the Huron and Iroquois Indians. Champlain fired into the Iroquois and killed two chieftains; as a result of which the Iroquois had no further dealings with the French. The Iroquois traded with the English and actually provided a buffer between French North America and the English colonies. The Hurons reflexively allied with the French.
In Spring, 1754, George Washington, then a twenty eight year old colonial in the British army, was sent to warn away French troops who had trespassed on land claimed by the British. A brief skirmish occurred, and a truce was called to discuss a resolution. When Washington turned his head, an Iroquois who was with him split the skull of the French commander and washed his hands in the man's brains; not that he had a quarrel with the Frenchmen; but rather because the French were allied with the Hurons, the Iroquois' hated enemies. The end result was a much larger battle and the opening salvoes of the Seven Years War in America.
Although the War technically ended in 1759 with the Battle of Quebec, there was still fighting for an additional four years as the Indians refused to stop. Pontiac led a group of Ottowa Indians on the side of the French in hopes of bringing about a pan-Indian revolt. In South Carolina, Cherokee Indians managed to push the frontier back 100 miles. The War ended with the Peace of Paris of 1763.
An indication of the importance of Indian relations is demonstrated by the Royal Proclamation of 1763 issued after the Peace of Paris. The Proclamation forbade English settlement west of the crest of the Appalachian Mountains in hopes of maintaining peaceful relations with the Indians who occupied that area.
An excellent source of further information is Fred Thompson's Crucible of War, which explains the origins and course of the Seven Years War and its relationship to the American Revolution.