The chief consensus, and the basic reason the delegates from the states convened when they did, was that the United States needed some other form of government than that which had been provided for in the Articles of Confederation. Although the major powers had recognized the independence of the US in the Treaty of Paris four years earlier (in 1783), no one had any way of knowing if this "experiment" in democracy, remote from the European homeland, was going to succeed. It was possible that it could collapse in the face of events like Shay's Rebellion, and moreover, the Articles did not provide enough (or any) mutual safeguards to the individual states in the case that their stability was endangered, either through domestic insurrections or the threats from the outside.
A disagreement existed over the issue of slavery. The question of whether a newly formed central government would have the power to free the slaves led to fear among the southern delegates, and then relief when the issue was (unfortunately for the future of the country) dropped, and the only provision enacted was that the slave trade would be ended twenty years in the future. The compromise that occurred was the famous (or infamous) three-fifths clause which provided that the enslaved people would be counted as that fraction of the free population for purposes of representation in the legislature. It's difficult to decide which side was the more hypocritical, the delegates who wished to deny the enslaved people their full status as part of the population, or those in the South who wished to keep the African Americans enslaved but still use their numbers in order to gain greater representation for themselves in Congress. Part of the compensatory attitude toward the South was the decision to place the future capital in the South (in Maryland), albeit in a middle position geographically, on the shores of the Potomac, between the northernmost and southernmost points of the country. The tragedy of the situation is that the whole issue of slavery was put on the back burner for some future generation to resolve.
Perhaps, however, the most fundamental (and controversial) question in the proceedings involved the type of government that would be created. A government divided among three equal branches—executive, legislative, and judicial—had never been quite attempted. The assumption of many would have been that a parliamentary system like that of Britain would be established. At this point today, 230 years later, it's an open question as to whether this tripartite system of government has been a success or a failure. The US government is genuinely unique. In Britain, for instance, there is no separation between the executive power wielded by the Prime Minister—in the eighteenth-century still the King, at least theoretically, and his minister(s)—and the majority in Parliament (of which the Prime Minister is a member): these together are the "government" and are in control of the entire system. By creating a structure in which the executive (the president) is not part of Congress (and in addition there is the third entity of the Supreme Court), the Constitutional Convention took the step of both establishing a strong central government for the US and simultaneously limiting the ability of any segment of it to control on its own the destiny of the nation.