Only Hobbits start in the Shire but you can go to all the starting areas for cheap at the stables in Michel Delving. Men and Man-bears start in Archet in Bree-land, all Elves start in Celondim in Ered Luin, and all dwarves also start in Ered Luin but in the north at Thorin's Hall.
When I start a new character I pretty much always complete all the starting areas before moving on. I'm always amused by how low stakes the Shire is in general.