The Wandering Days was the term used by Hobbits to refer to the centuries-long period of their westward migration from their races' ancestral home somewhere in the wild East (probably in the upper Vales of Anduin).
The oral history and memory of the Hobbits did not extend further back than their "Wandering Days", with their exact origins having been lost. So the first known historical fact is that they were alerted by the rise of Dol Guldur, so they drifted westward. Around T.A. 1050 many Harfoots settled in Eriador and about 1150 the Fallohides followed, while many Stoors who emerged from the Redhorn Pass settled near Tharbad in Dunland, specifically the Angle. In 1300 the Kingdom of Angmar was founded, so the northern Hobbits had to flee from further westwards, reaching Bree. Some Stoors (in 1356) returned to the Gladden. The others continued westward and their first settlements were in Bree and Staddle.
Eventually, in the year T.A. 1601 a large population of Bree-hobbits was granted permission by King Argeleb II, one of the last few Kings of Arthedain, to colonize the region beyond the Brandywine, and were later joined by the Stoors of Dunland. This would become The Shire, the main homeland of most of Hobbit-kind.