There are two known writing systems used on the island of Crete during the Minoan period (27th to 15th century BC). One is called Linear A and the other is Cretan hieroglyphics.

In the later Mycenaean period, a script derived from Linear A was used. It is called Linear B. It was deciphered by Alice Kober and Michael Ventris in the 1940's and 50's. It turns out that Linear B was used to write the Greek language, a task to which it was poorly suited. It was a syllabary, meaning that each character represents a syllable rather than a phoneme. There were around 200 characters.

Linear A probably was used to write a different language than Greek. Even though the symbols are mostly the same, it doesn't make sense in Greek if you try to use the same phonetic interpretation as Linear B. It is likely that it represents an older language, now extinct, that was once spoken on Crete.

Source: Language and the Ancient Greeks, Richard E McDorman

I also recommend The Decipherment of Linear B by John Chadwick

Were there any early interactions with the Mesopotamian civilizations that could have lead to an import of the Cuneiform script? Or was it seen as archaic by that period?

The bronze age Greeks (before ~1200 BC) were the Minoans and the Mycenaens. These two cultures used linear A and linear B scripts, respectively. Linear A developed from Creten hieroglyphics and was superseded by B from ~1500. Linear B died out during the Greek Dark Ages (~1200-800) after Mycenean civilization collapsed.

Of these two scripts, Linear B has been deciphered. Phoenician characters were being used by the 8th century.