As the first President of the United States, almost all of Washington's actions shaped the Presidency. When he took the oath of office, he wore a civilian suit but also wore a sword, to indicate that the Commander in chief of the armed forces was a civilian. Since that time, no President of the United States has worn a uniform while in office.
Washington also created the first four cabinet positions, three of which remain in effect to this day: Secretary of State, Secretary of the Treasury, and Attorney General. He also was responsible for the position of Secretary of War, which has been succeeded more recently by the Secretary of Defense.
Additionally, when John Jay, then Chief Justice, negotiated a Treaty of Friendship with Great Britain which offended the French, the House of Representatives demanded that information on the treaty be sent to it for review. Washington refused, noting that Treaties were the particular business of the President with the advice and consent of the Senate, not the House. He thereby set the precedent for executive privilege.
Finally, by refusing to seek a third term in office, Washington set the precedent of a two term limit for the presidency. This precedent was broken only once, by Franklin D. Roosevelt; but since has been made a part of the Constitution.