The Baratheons are from Storm’s End in the Stormlands, on the eastern coast of Westeros.

They are formally known as House Baratheon of Storm’s End, one of the Great Houses of Westeros, ruling the Stormlands as Lords Paramount. The house itself was founded when Orys Baratheon, a commander for Aegon the Conqueror, defeated the last Storm King, took his castle of Storm’s End, and was granted that ancient fortress and the surrounding storm-battered region as his seat.

So, if you’re just looking for the quick scoop on “where” they’re from:
They hail from Storm’s End, in the Stormlands of Westeros.

Information gathered from public forums or data available on the internet and portrayed here.