Here's a summary of what she says:
* Secret 1: She mentions "a place where the old man rests," hinting at Cinnabar Island and the Fossil Restoration mechanic.
* Secret 2: She tells you about a "powerful Trainer who lives in a mansion on a mountain." This refers to Giovanni and Silph Co. in Saffron City.
* Secret 3: She talks about a "special location where you can find a rare Pokemon," hinting at Cerulean Cave, a place with many rare and powerful Pokemon.
Note: These "secrets" aren't really hidden information, but rather hints that help you progress through the game. They're like "easter eggs" meant to be discovered and enjoyed by observant players.
It's easy to mistake her for a lady, but she's actually a very skilled Pokemon trainer who's been around for a long time!