Because the Constitution was a brand new document, the early years were a time of testing the waters more or less to see what would happen, and this took no longer than the election of John Adams, the second president. By the time Adams was elected, followers of Alexander Hamilton and Thomas Jefferson had morphed into something like political parties; Hamilton's people favored strengthening the central government through a relaxed interpretation of the new Constitution ("loose constructionists") and focusing on economic growth, while Jefferson's were more focused on developing the nation's agriculture and reining in the central government through a strict constructionist view of the Constitution.
Relations with France and England constituted two thorns in the side of the new nation; attempting to warm-up frosty relations with France in the years after the Constitution was ratified resulted in the "XYZ Affair" where three American diplomats, dispatched to work out a peace agreement, were asked to pay bribes (not an unusual custom in European politics at the time). Two of the three diplomats became offended and left, although the remaining third gentleman did remain and manage to conclude an agreement. The real test of the new Constitution, however, emerged as a mild consequence to the "XYZ Affair" when Adams succeeded in getting the Federalist-dominated Congress to pass the "Alien and Sedition Acts", the purpose of which was twofold. First, it was an attempt to minimize the possiblity of French immigrants creating unrest within the United States should war break out, and secondly, it was intended to help Adams consolidate his power, which was weakened by the fact that his vice-president, Thomas Jefferson, was a member of the rapidly forming opposition party. Thus, the first real test of the Constitution unfolded.
In a stunning precursor the the Civil War that would erupt on 1861, James Madison and Thomas Jefferson drafted the Virginia-Kentucky Resolutions, which stated that a state or states could nullify a federal law that said states deemed unconstitional. This was an interesting idea for a couple of reasons, not the least of which was that it gave the states, rather than the Supreme Court, the power of judicial review. This was the beginning of years of debate over the authority, or lack thereof, of the central government, as perceived by the two new political parties.