Now that the Revolutionary War had been fought, how did the framers and revolutionaries go about forging and forming a national republic/new nation between 1776 and 1789?

As soon as the colonies agreed to declare independence, each set about establishing a state government. The constitutions that were written and established in the various states varied in form—Pennsylvania's for example, was quite democratic—but they generally established governments based on the principles of separation of powers and representation, a set of beliefs sometimes called republicanism. These governments struggled with fiscal issues in particular, as the Revolutionary War placed a tremendous strain on scarce resources. The states also struggled to agree on a national government, establishing a loose organization known as the Articles of Confederation, which did not go into effect until 1781, when the war was all but over. This government reflected the fear, shared by many Americans, of excessive centralized power. The congress it created was very weak, lacking the power to tax, and there was no centralized executive power. Fiscal crises at the state level led to popular unrest, and many of the Founders feared that state governments were too democratic, and the national government too weak, to deal with the issues confronting the nation. The Constitutional Convention of 1787 was intended to address these issues, and it established a national republic that curtailed the powers of the states while allowing for limited participation in government by the people. Mainly, it sought to deal with the challenges of the postwar era by placing expansive powers in the hands of the central government. It was the founding document for the new nation that was forged out of the chaos of the post-Revolutionary War period.
https://www.archives.gov/founding-docs