The problems faced by the new republic went on from 1776 until the Constitution was ratified in 1787. These problems included a nonexistent executive branch under the Articles of Confederation. The Constitution addressed this issue by creating a President and an executive branch.

In addition, under the Articles of Confederation, the federal government had a hard time raising monies from states to fight the Revolutionary War. To address this issue, the Constitution created a legislative branch, or Congress, that had the ability to raise money through taxation.

Also, during the early republic, there was an ongoing battle between the large and small states over power and representation. To address this issue, the Constitution created a bicameral legislature, meaning a Congress with two houses. Representation in one house was determined by population, while in the other house, each state had the same number of representatives. This was a compromise struck between large and small states.

By 1789, the Founding Fathers recognized the Articles of Confederation were not sufficiently strong to hold the union together.  Some of the problems included:

1.  Each state had their own laws and the Constitution established federal supremacy over state laws.

2.  Each state was coining its own money and the Constitution provided that only the national government could coin money.

3.  The Articles relied on voluntary support from the states for operating funds and the Constitution established the ability of the national government to levy taxes.

4.  While leaving each state with its militia, the national government assumed the responsibility to protest the states from foreign and domestic threats.

First of all, please note that the Constitution did not get written until 1787 and was not ratified until 1788.  Many of the challenges faced by the new United States in 1776 (such as how to win independence) were not reflected in the Constitution.  In my answer to this question, I will touch on some of the main challenges that faced the US and that were reflected in the Constitution, even if those challenges did not arise until after 1776.

One major challenge was for the US to defend itself.  To do this, it needed a strong central government that could levy taxes and pay for a military.  The Articles of Confederation did not provide this.  The Constitution was written to create a stronger national government with the power to levy taxes.  This was partly in response to the need for national defense.

A second major challenge was the challenge of keeping the slave and free states together.  This was reflected in the Constitution by the 3/5 Compromise and the compromise that protected the slave trade for 20 years. 

A third challenge was to keep the big and small states together when they could not agree on how much representation they should each have in Congress.  This was reflected in the Constitution by the Great Compromise in which all states had equal representation in the Senate while the big states had more representation in the House of Representatives.

A fourth challenge had to do with trade between the states.  Before the Constitution was written, states were engaging in trade wars with one another.  This reduced the size of the US market and made the country’s economy weaker.  To combat this problem, the Constitution gave the Congress the power to regulate trade between the states, thus prohibiting the states from raising trade barriers against one another.

All of these were important challenges that the early United States faced and that were reflected in the Constitution.