The Baratheons started with Robert. He was a drunk who loved war and whores. His “official” kids (Joffrey, Myrcella, Tommen) were Lannisters. His brothers Stannis and Renard fought for the throne. Stannis burned his daughter alive. Yikes. Renly got killed by a shadow baby. Yes, a shadow baby.
The Baratheon line is basically extinct. Gendry, Robert’s bastard, becomes lord of Storm’s End. So the stag lives on, but barely.