The Constitution of the United States came into effect when New Hampshire became the Ninth State to ratify it. (Article VII - "The Ratification of the Conventions of nine States, shall be sufficient for the Establishment of this Constitution between the States so ratifying the Same."). That occured on June 21st 1788. However the Institutions came into existence on 4th March 1789. That was the date set for the opening of the 1st Congress.
There is a detailed description of the events of that day in Robert Remini's excellent history of the House of Representatives - "The House". I would thoroughly recommend the book to anyone interrested in American History - but particularly the history of the House of Representatives. It is both a good read (and I've read it straight through on a number of occasions), and an excellent research resource (of which I make frequent use).
The day was one of great ceremony. Flags flew around the city of New York, which was host to the new Government. The centre of attention was Federal Hall - which wasw at that time New York's City Hall. It had been converted for the nation's use by Pierre-Charles L'Enfant. Sadly, while the crowds were there, few elected Congressmen were. Travel had been hampered by the poor states of many "roads" - many "mired in mud, riddled with potholes or washed away by floodwaters". Ice-packed rivers also presented an obstacle. It was only on 1st April that the number needed for business - a quorum - was reached.