Looking at theof NYCB, there are other interesting things to see besides Swan Lake. For a newbie "leaning in" to Ballet, these might be the most fun to see. Not sure if you're ready for the modern / astringent stuff, so I'm sticking with the accessible programs:
October 5: 3 short stories: La Sonnambula, Prodigal Son, Slaughter on Tenth Avenue
January 22, 23 25, 28, 30, February 2, May 23, 24, 25: Jewels
January 24, February 1, 6, 9: Dances at a Gathering, Union JackFebruary 14, 15, 16,17, 22, 23: CoppeliaMay 6, 7, 10, 11, 12: Raymonda, Tin Soldier, Le Tombeau, Symphony in CMay 14, 15, 17, 20: Davidsbündlertänze”, Union JackJune 3,4,5,6,7,8: Midsummer Night's Dream (the 2nd act is a bunch of divertissements without any storyline, but the ppd is worth it)
I wouldn't send a newbie to Davidsbundlertanze. I love it, but many people find it tedious. I would send a newbie to a Black & White program (or a program that contained at least one B&W ballet) because I think that it is essential viewing to have an understanding of Balanchine's neoclassical ballets.