For favorite operas, it’s really hard to pin it down to one, but among them would be La Boheme, Il Trovatore, Rigoletto (all three of them tragic operas), and, for a lot of very melodic fun, the comic opera Die Fledermaus. Boheme and Fledermaus are two of the ones I would recommend to an inexperienced operagoer because they are so melody-rich and so easy to follow (the storyline actually has some kind of appreciable logic to it), and they are typically visually delightful as well.
As for ballet, after all these years I am still a pushover for Swan Lake, but I also love Prokofieff’s Romeo and Juliet.
For wonderful comic light opera (often referred to as operetta; there’s plenty of spoken dialogue), those of Gilbert & Sullivan are in a class by themselves. The “big three” are The Mikado, H.M.S. Pinafore, and The Pirates of Penzance. I also love the others, especially Iolanthe, Patience, and Princess Ida. I have a special affection for The Yeomen of the Guard, which is the only one that doesn’t end happily for everyone.