The Revolutionary War is composed many separate battles, won by both the American armies and the British armies. The first "battle" of the war was the Battle of Bunker and Breed's Hill. The next major battle was the Battle of Long Island. Two of the most important battles were the Battles of Trenton and Princeton. The British taking control of Upper New York was supposed to be a big event, but the favors turned to the Patriots.
As for fighting in the South, things were quite different, especially with the the Battle at Charleston. One of the most important battles in the South was the Battle of Yorktown.