I would say that President Washington faced several challenging elements in being the first President of the new nation. One challenge came from the national debt. Fighting and for and gaining independence did not come cheap and the amount of debt that the new nation faced was staggering. He was able to navigate the challenges of the role of the federal government with such raging polarities in his cabinet with the likes of Alexander Hamilton, a proponent of an intervening government, with Thomas Jefferson, an advocate for a less intrusive form of government. With challenges abroad, Washington was also facing challenge with the role of America in the new world order. Would America assert its new found might in other problems all over the world or would the nation remain more inwardly drawn in this regard? Domestic uprisings like the Whiskey Rebellion also proved to be a challenge that Washington faced down with his use of the army to quell such an insurrection.
Washington was responsible for getting the country off on a firm foundation after the new government was formed. One of his first obstacles was the debate between Jefferson and Hamilton over the Bank of the United States. There was also the issue of funding the debt of the states from the Revolution, which Hamilton proposed and Washington opposed. Then, of course, there was the Whiskey Rebellion, caused by opposition to Hamilton's tax on Whiskey. Washington showed his magnanimity by commuting the sentences of those convicted in the Rebellion. He issued his declaration of Neutrality when most of his cabinet wanted the United States to intervene in the ongoing war between Britain and France, and again showed his diplomatic skills during the Citizen Genet affair; when he easily could have had Genet deported to certain death; but instead allowed him to remain in the United States.
The major issue faced by George Washington was president of the new United States was to get the country off on a start to democracy. It would have been very easy for the country to become something of a monarchy and there were those (like Alexander Hamilton) who would have welcomed this. By resisting this temptation (to become king) Washington set the US on the path to becoming a true democracy.
When Washington became president, there was little precedent for a country that did not have a ruler. In some way, Washington was going to need to govern the country well yet, at the same time, keep from having the political system revolve around the president as an individual. This is a challenge that many other new countries have been unable to solve. Because Washington was able to solve the problem, the US became a stable democracy instead of becoming a country ruled by a series of strongmen or dictators.