when did they start their journey to this place?
End of July 1620
"Mayflower" voyage
"Speedwell" was re-rigged with larger masts before leaving Holland and setting out to meet "Mayflower" in Southampton, England, around the end of July 1620
which group of english immigrants first settled in plymouth in 1620 Puritan Separatists
1620 They  obtained a land patent from the Plymouth Company in June 1619. 1620
Plymouth Colony Plymouth Colony Pilgrims (Plymouth Colony)
Introduction Origin Voyage
Plymouth Colony (sometimes Plimouth) was an English colonial venture in America from 1620 to 1691 at a location that had previously been surveyed and named by Captain John Smith. 
The congregation obtained a land patent from the Plymouth Company in June 1619.  It departed Delfshaven in July 1620 with the Leiden colonists, after a canal ride from Leyden of about seven hours. It reached Southampton, Hampshire, and met with the "Mayflower" and the additional colonists hired by the investors. With final arrangements made, the two vessels set out on August 5 (Old Style)/August 15 (New Style).
wiki:19464319
Plymouth Colony [SEP] "Mayflower" voyage
"Speedwell" was re-rigged with larger masts before leaving Holland and setting out to meet "Mayflower" in Southampton, England, around the end of July 1620. The Mayflower was purchased in London. The original captains were Captain Reynolds for "Speedwell" and Captain Christopher Jones for "Mayflower". Other passengers joined the group in Southampton, including William Brewster, who had been in hiding for the better part of a year, and a group of people known to the Leiden congregation as "The Strangers." This group was largely made up of people recruited by the Merchant Adventurers to provide practical assistance to the colony and additional hands to work for the colony's ventures.