When Melba first stepped ashore on American soil late November 1893, her first impression was:

Cobblestones and jolting cabs! That is my first memory of New York, where I landed in the winter of 1893.(1)

Another thing Melba quickly noticed was the importance of opera singers to the American people:

In London, if an artist made a great success, he or she was received on a footing of absolute equality with the most “exalted “ people in the Capital. Not so in New York. An artist was an artist, and although she might be the subject of amazing hospitality, though innumerable kindnesses might be showered upon her, there was always a subtle difference between her and the rest of society. (2)

Melba made her operatic debut before American audiences on December 4, 1893 at the Metropolitan Opera House, New York. However, her debut was less than ideal – firstly, she had picked Lucia di Lammermoor as her debut opera which the Americans felt was Madame Patti’s opera. Also:

…. on the eve of my appearance, one of the principals suddenly fell a victim to laryngitis, and I had to act with an Enrico who had not even once rehearsed the part. In addition, there had been no Grand Opera in New York the last season, owing to the fire at the Metropolitan, which had necessitated extensive building alterations. A new house, a new audience, a new country—it was not unnatural that I felt nervous. (3)

How I conquered that nervousness is, I think, best told in the words of the critics:
“The temptation is strong to say that in vocal equipment she surpasses all her rivals. In finish of vocalization she is the finest example heard on the local stage since Sembrich made her debut here ten years ago. Her voice is charmingly fresh and exquisitely beautiful, and the tone production is more natural and more spontaneous than that of the marvellous woman (Patti) who so long upheld the standard of bel canto throughout the world.”
New York Tribune, Dec. 5th, 1893.(4)

Melba also felt the acoustics of the Metropolitan were not as good as Covent Garden, the Auditorium in Chicago or La Scala.

However, she praised the knowledgeable audiences which were slow to accept her:

American audiences are among the most intelligent in the world. I do not know how much mere dollars have to do with this, for one cannot help remembering that they have always had the money to pay for what they wanted, and have always been able to afford the best. The fact remains that they know what is good and that they will never put up with a “second best.” And they judge for themselves without waiting for the critics. (5)

To Melba it was not until she sang in Romeo and Juliette on January 19, 1894 that she felt she had established herself with American audiences.

Her first season at the Metropolitan stretched from December to the end of April, 1894 and included opera and concerts (6):
December 4: Debut as Lucia in Lucia di Lammermoor.
December 6: Ophelia in Hamlet.
December 11 & 22: Nedda in I Pagliacci.
December 29: Gilda in Rigoletto.
January 14, 1894: Concert included ‘Air du Rossignol‘ (Tenserioso’), Handel; Tosti’s ‘Goodbye’.
January 12: Semiramide in Semiramide.
January 19: Juliette in Romeo et Juliette.
January 21: Concert included the Mad Scene, Lucia di Lammermoor, Waltz, Romeo et Juliette.
January 29: Elizabeth in Tannhauser.
February 6: as Elsa in Lohengrin.
February 10: Juliette in Romeo et Juliette.
February 14 to 16: Gala concert featuring Act 4, Rigoletto and Act 2, Hamlet.
February 18: Concert and Melba performed ‘Sweet bird’ (‘Penserioso’), Handel; Duet (unspecified), Handel; ‘Se saran rose’, Arditi.
February 23: Marguerite in Faust.
February 25: Concert featuring ‘Ballatella’, Pagliacci; ‘Caro nome’, Rigoletto.
April 16: Marguerite in Faust.
April 20: Semiramide in Semiramide.
April 22: Concert featured ‘Sweet bird’; ‘Bel. raggio’, Semiramide.
April 26: Lucia in Lucia di Lammermoor.
April (undated): Gala concert featured the Mad Scene from Hamlet.

References:

(1) N. Melba, Melodies and Memories, Thornton, 1925, pgs 121.
(2) Op. cit., pg122.
(3) Op. cit., pg123.
(4) Op. cit., pg123.
(5) Op. cit., pg124.
(6) T. Radic, Melba The Voice of Australia, The Macmillan Company of Australia Pty Ltd, Melbourne, 1986, pgs 189-191.