The quick and easy way out of this would be simply to answer, “Well it depends who you are!” Much like today, Americans of the early republic had many gripes with the American government and with each other. What follows is a list of some of the largest complaints, though these certainly do not cover every single complaint against the government.
Most importantly, let’s remember that the year 1800 was an election year. Although Americans that lived through 2016’s election might contend that modern elections are particularly brutal, elections of the eighteenth and nineteenth centuries could be fierce as well and 1800 was among the first national elections to be bitterly contested. I don’t want to dive too heavily into the election of 1800 in order to dedicate time to other subjects; however, I would recommend looking at /u/sunnymetoaddict's if you’re interested in more detail. Briefly, 1800 was a rematch of the 1796 election that pitted pro-French anti-centralization Democratic Republicans led by Thomas Jefferson and Aaron Burr against pro-English and pro-centralization Federalists led by John Adams and Charles Pinckney. One of the most intriguing issues of the campaign was the Democratic Republican claim that Federalist support of the “Alien and Sedition Acts” was subverting principles of the American Constitution. This belief was also held by a number of Federalists who followed Alexander Hamilton. Briefly, Adams and followers believed foreign threats caused by the European rivalry between France and England were jeopardizing the young nation and so passed legislation that gave the President executive power to censor certain individuals deemed threatening. Jefferson and his followers argued this was a blatant violation of the Bill of Rights. From /u/Furious_Georgee Read more here. When the election finally came, the process ran into a serious problem in the process. At the time, there was no vice-president running position and electors in the Electoral College passed two votes for president. When the votes were totaled, the leader became president and the man who received the next most votes became vice-president. The Democratic Republicans had a plan that one member would cast only one vote for Jefferson and so ensure that Jefferson became president and Burr vice-president. In typical political fashion, this plan was not followed and there was a tie for President between Burr and Jefferson. In case of a tie, the House of Representative were to cast ballots to determine the winner. The problem was, the Federalist states (primarily in New England) were determined to block Jefferson and voted for Burr. Only after the 36th vote, and Jefferson’s promise to keep Hamilton’s financial system was Jefferson elected by the House of Representatives. Needless to say, many Americans criticized the electoral process and soon afterward the 12th Amendment was passed to prevent that problem from arising again.
Closely related, the events occurring in Europe in 1800 greatly affected Americans. Most importantly the ramifications of the French Revolution and the Quasi War between France and England was hurting American shipping. Although the French had allied with the Americans during their revolution, following the dramatic years of the 1790s French Revolution, even pro-French politicians like Jefferson were wary of supporting the bloodshed. Further complicating matters, Napoleon (and others) solidified the end of the republic and began building a dictatorship in the early months of 1800. New Englanders, heavily involved in shipping and European trade, supported the British despite having only separated from the Empire in the last quarter-century. Meanwhile, the more agrarian south was suspicious of growing industrialization of the North and its ties to English textiles despite their reliance on the infant textile industry for consumption of their emerging cotton exports. While this rivalry would not come to ahead until 1812, division among Americans was apparent even before 1800 because it hit upon the long-standing problem of centralization. Supporters of the English recognized that the centralization of power in a federal system offered the best protection of industry, trade, and the nation through military while those who were wary argued that a federal government inherently restricted freedoms that were better preserved in local government. While the pro-British/pro-French debate was not entirely about centralization, it often served a very thin veil.
Surprise, surprise, slavery was also an issue in 1800. The beginning of the opportunity to ban the international slave trade was cemented in the Constitution. According to the law, no federal law could prohibit the trade prior to January 1, 1808. However, since the 1790s states had moved to get rid of the peculiar institution. For example, Massachusetts would not formally ban slavery until 1865 and the 13th Amendment; however, no slaves were recorded in their 1790 census because the state had tightened its indentured servants laws proscribing what could and could not be done to people in bondage. However, in 1800 slavery was not a strictly North versus South debate, nor was in 1860 for that matter either. Rhode Island, New York, New Jersey, and Pennsylvania had many slaves especially in the urban centers. By 1800, however, many American abolitionists had begun movements to band the practice and some Christians sects (mostly in New England and Pennsylvania) began isolating congregation members that owned slaves. The government was criticized for both not protecting slave-owner’s rights and for not limiting slavery.
Finally, at least for this piece, American boundaries in the west and the “Indian Problem” had caused colonists to grow frustrated with the British in 1763 and increasing movement westward was causing a similar problem in 1800. Large Indian Territories were located in the Great Lakes region just west of the “Northwest Territory” and along the Gulf Coast. Population increase and the growing cultivation of cotton, which would one day become “King Cotton” pushed Americans west in search of land. Land west of Mississippi had been owned by Spain and by 1800 France had regained “Louisiana” but Americans were already beginning to stream across the river into western lands of which the French had little imperial control. Ultimately, Jefferson would purchase Louisiana in 1803, but even in 1800 the question about what to do with native peoples and how the western states should be governed drew critiques from all sides. The growth of slavery, indigenous status in the new nation, and how land would be distributed was complex and many frontier people complained the costal elite paid little attention to their needs.
I would argue these are among the top issues that drew grievances from the American people in 1800 but there were of course many more! I chose these because over the next half-century, these issues would dominate the political atmosphere of the United States and set the tone for the major reckoning that was the Civil War.
A Few Sources:
Hofstader The Idea of a Party System (1970)
Elkins and McKitric, The Age of Federalism (1995)
Beckert, Empire of Cotton (2014)
Taylor, The Civil War of 1812 (2010)