I'm writing a novel about ancient Greeks settling in Italy Sicily and Libya. How would they call these areas? The novel also involves them going to Delphi to ask for advice about settling there. How would a city go about asking for such a divination? And would a typical divination of such nature be delivered?
I'm answering only to the geography part:
Generally speaking one should consult firstly the geographers, Strabo of the 1st c BCE and Claudius Ptolemy 2nd c CE for these matters. However, names can be found of course in other earlier sources.
Regarding the names you've asked [Italy, Sicily, Libya and Magna Graecia], in brief they are as follows:
The names of Italy and Sicily [Ἰταλίην τε καὶ Σικελίην] are surely existing since the years of Herodotus of the 5th c. BCE [Hdt. 1.24.1].
Regarding Sicily: poet Homer of the 8th c. BCE is mentioning an island called Thrinacia [Θρινακίην; Od. 11,107, 12.127], which really later Strabo [Strab. 6.2.1] is identifying as an alternate spelling of the older name Trinacria that was standing for the island of Sicily; the name 'Trinacria' is meaning 3 ends cause of its triangular shape. Thucydides [Thuc. 6.1-2] intermediately in the 4th c. BCE is also mentioning the name of Trinacria for Sicily as older, along with Sicania [from the Sicaneans, the claimed first settlers of the island, coming from Iberia].
The name of Libya [Λιβύη] is another case. It's mentioned already since Homer but in an unclear way. In Herodotus is almost clearly standing for the continent of Africa, next to Europe and Asia [4.42]. Naming that is repeated by Strabo, too [eg. Strab. 2.5.26], where is meant the north Africa. What is called today the country of Libya, was mostly called then Cyrenaica, by the colony of Cyrene in north-eastern Libya; already since Herodotus [Hdt. 4.199], repeated by Strabo [Str. 17].
The name of Magna Graecia stands for Greater Greece [Μεγάλη Ἑλλάδα] and as far as I know it appears first time in Polybius of the 2nd c. BCE, when speaking about Pythagorean schools ther. The term is repeated by Strabo, too [Strab. 6.1.2], who nevertheless is speaking of greek movements in the area since the Trojan war. Later Iamblichus of the 3rd c. CE attributed this naming to Pythagoras, or better to his later students cause of Pythagoras [Iamblichus - De vita Pythagorica 6.30 & 29.166].