What were some of George Washington's issues when he was president, and how did he solve them?

As the first president, nearly every issue George Washington dealt with required him to invent the solutions, as no precedents existed and the the practical responsibilities of the presidency were not well established. When he took office, the only actual part of the federal government that existed were those offices created by the Constitution, so the United States had a Congress but no agencies or departments or even a functional court system.

On of the first things he did was establish the Supreme Court through the Judiciary Act of 1789. The six-member court, in theory, would mediate conflicts between the states and oversee the lower federal courts.

Washington also established the first cabinet, surrounding himself with many of the more influential members of the Revolution. Within just a short time, his advisors split largely into two camps. One one side, Alexander Hamilton pushed for a stronger federal government with a centralized bank and a permanent army. On the other side, Jefferson wanted to limit the power of the federal government. Each cabinet member oversaw a new department, like the ones we have today such as Treasury, War (now Defense), etc. These departments, in turn, became the institutions of governance in the young nation. 

Washington needed to create revenue for the government to pay off debts incurred from the Revolution, so, siding with Hamilton, he secured new taxes on a number of luxury goods. The most unpopular, the Whiskey Tax, sparked an event that became known as the Whiskey Rebellion. Distillers in Pennsylvania revolted, and Washington assumed command of a federal army and met the rebels, ready for combat, but they backed down. His leadership helped legitimize the authority of the new federal government. 

Finally, the political landscape of Europe changed rapidly during his presidency, as the French Revolution ended the monarchy and threatened to spread throughout Europe. France had been the major ally of the American Revolutionaries and many Americans viewed the French Revolution, at least initially, as an extension of their own fight for liberty. Jefferson was a particularly vocal supporter. Hamilton and many in his camp opposed any American support of the Revolution in France and favored moving the United States closer to England, repairing relations and promoting commercial ties. Washington issued a proclamation of neutrality, which bought the Americans some time, but also set up his successors for a much more serious contest between French and British alliances as Hamilton and Adams vied for power with Jefferson in the coming decade.