What were the most serious problems facing the new nation when George Washington became president?
There were two main issues facing the United States when Washington became president (as opposed to later in his terms when other issues arose). They were the need to unite the country and the need to create a strong economy.
When Washington took over as president, the various states were not used to being part of one country with a strong central government. There were many Antifederalists who were not at all sure they liked the new system. This problem of potential disunity was one of the most important that the country faced.
The country was also relatively weak in economic terms. The country had removed itself from the British Empire and thereby severed some of its most important economic ties. The country had to find a way to become prosperous again.