The foundation myth of Carthage, though only preserved in late sources, tells us that a certain "Dido" fled Phoenicia in ships laden with her slain husband's hoard of gold, and that she settled in North Africa to found a new city (Carthage, Kardt-Hadash, means "New City" in Phoenician). We have no idea if this story has any sort of historical veracity to it, but it does bring up a valid point: there were many reasons that people(s) were moving around in that period, and not all of them concerned "colonies" and "homelands." We don't know for sure what exactly the Phoenicians were up to in the 9th and 8th centuries BCE, but I am fairly confident that it looked very different than British colonial behaviors two millennia later.

Personally I think the Phoenicians went west in search of raw materials, especially precious metals. The peoples of the Levant had inherited a long tradition of artisan metalworking, and were the masters of turning raw silver and gold into elaborate pieces for resale. In particular, the Phoenicians were very good at granulation techniques by which they decorated pieces with tiny dollops of molten gold. In pursuit of metals, then, they went ... where the sources of raw metal were. That meant Cyprus, initially, where they had a major settlement at Kition. Beyond Cyprus, that meant a few juicy spots in Greece, all of which had occupants who probably proved to be less than cooperative. They probably then went further afield, drawing on knowledge of the central and western Med. that stretched back into the Bronze Age. That meant southwestern Spain, where they founded a port at Gadir (Cadiz). That meant Andalusia, Sardinia, Sicily, and northern central Italy. In all of these places, there is pretty clear evidence that they interacted with "the locals" (Iron Age Iberian tribes; Etruscans; Sicels; etc) in order to work out deals for extraction. In SW Spain, it was via Gadir and via smaller, "emporion"-style enclaves like that of Castillo de Dona Blanca. In Andalusia, it was a string of small, coastal settlements which seemed to serve as an interface for the extraction of materials from the unexplored interior (Toscanos, e.g.). In Sardinia, there were several large settlements from early on in the 8th century, including Tharros (from which the image linked above comes). In Italy, there are no adjacent settlements (that we've found), so they must have relied on a different model: routine visits to Etruscan cities, for instance, or perhaps use of a common meeting ground like Pithekoussai(?).

Recent scholarship suggests that Carthage, from the get-go, was a sort of centralized clearing house for the movement of materials back and forth, and rather quickly outgrew the importance of the "homeland" as a hub for the reception, working, and re-export of Phoenician products. We can trace the creative output of Carthage workshops and find "Carthaginian" vs "Phoenician" wares in the west as early as the end of the 8th century. We don't know what relationships existed between the west and the Levant, or Carthage and Tyre/Sidon, though by the 6th century it didn't matter, because the Phoenician "homeland" had been conquered and Carthage was left as the new de facto capital of Phoenician (now "Punic") interests.