For one thing, the 1788 elections were held in 1789.
The following quote is from p. 166 of Sol Bloom, The Story of the Constitution, United States Constitution Sesquicentennial Commission, House Office Building, Washington, D.C., 1937.
Q. When did the United States government go into operation under the Constitution?
A. The Constitution became binding upon nine States by the ratification of the ninth State, New Hampshire, June 21, 1788. Notice of this ratification was received by Congress on July 2, 1788. On September 13, 1788, Congress adopted a resolution declaring that electors should be appointed in the ratifying States on the first Wednesday in January, 1789; that the electors vote for President on the first Wednesday in February, 1789; and that "the first Wednesday in March next [March 4, 1789] be the time and the present seat of Congress the place for commencing proceedings under the said constitution." The Convention had also suggested "that after such Publication the Electors should be appointed, and the Senators and Representatives elected." The Constitution left with the States the control over the election of congressmen, and Congress said nothing about this in its resolution; but the States proceeded to provide for it as well as for the appointment of electors. On March 3, 1789, the old Confederation went out of existence and on March 4 the new government of the United States began legally to function, according to a decision of the Supreme Court of the United States (Owings v. Speed, 5 Wheat. 420); however, it had no practical existence until April 6, when first the presence of quorums in both Houses permitted organization of Congress. On April 30, 1789, George Washington was inaugurated as President of the United States, so on that date the executive branch of the government under the Constitution became operative. But it was not until February 2, 1790, that the Supreme Court, as head of the third branch of the government, organized and held its first session; so that is the date when our government under the Constitution became fully operative.
Thus, the old congress set the date for the choosing of electors and date for the electors to vote, all other details being left to the states. Note that, while representatives must be chosen by the people, not so for electors. Article II, Section 1 of the Constitution says:
Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, [. . .]
So it's up to each state legislature to decide whether to appoint electors directly, or have them chosen by popular vote, or in some other manner. According to Wikipedia, of the 11 states in the Union at the time of the election:
- in four states (CT, GA, NJ, SC) electors were chosen by the legislature;
- in four states (DE, MD, PA, VA), electors were chosen by popular vote, either at large or by district;
- two states (MA, NH) used a hybrid system;
- one state (NY) failed to choose electors.