A slightly different question from "what was the world's first writing system?" is "what was the world's first alphabet?" Barry B. Powell in his 2009 publication Writing: Theory and History of the Technology of Civilization provides invaluable insight into this question.
The Word Alphabet
West Semitic people from the eastern coast of the Mediterranean (where Phoenician and Hebrew groups lived) are usually credited with developing the world's first alphabet. It was a short, 22-character list with (1)names and (2)a fixed order for characters that could (3)easily be memorized. This "alphabet" was spread by Phoenician traders and then modified by the inclusion of vowels, by the Greeks, whose first 2 letters, alpha and beta were put together to form the name "alphabet."
In Hebrew, the first two letters of the abecedary (as in A-B-C) are, likewise, aleph and bet, but unlike the Greek letters, the Semitic "alphabet" lacked vowels: Aleph wasn't an /a/. In Egypt, too, writing has been found that uses only consonants. Egypt could be named as the nation with the first alphabet were the provision of vowels considered unnecessary.
Barry B. Powell says it is a misnomer to refer to the Semitic abecedary as an alphabet. Instead, he says the first alphabet is the Greek revision of Semitic syllabic writing. That is, an alphabet requires symbols for vowels. Without vowels, consonants can't be pronounced, so only partial information on how to read a passage is provided by just the consonants.
Poetry as Inspiration for the AlphabetIf the vowels are dropped from English sentences, while the consonants remain in their correct position with respect to the other consonants, literate, native English speakers can usually still understand. For example, the following sentence:
Mst ppl wlk.should be understood as:
Most people walk.This may be opaque to someone not raised on English, perhaps especially if his native language is written without an alphabet. The first line of the Iliad in the same abbreviated form is unrecognizable:
MNN D T PLD KLS
MENIN AEIDE THEA PELEIADEO AKHILEOS
Powell attributes the Greek invention of the first real alphabet to the need for vowels to transcribe the meter (dactylic hexameters) of the great epics, Iliad and Odyssey, attributed to Homer, and the works of Hesiod.
Greek Modification of the Phoenician Symbols
Although it is conventional to refer to the introduction of the vowels by the Greeks as an "addition" to the 22 consonants, Powell explains that some unknown Greek reinterpreted 5 of the Semitic signs as vowels, whose presence were required, in conjunction with any of the other, consonantal signs.
Thus, the unknown Greek created the first alphabet. Powell says this was not a gradual process, but the invention of an individual. Powell is a Classical scholar with publications in Homer and mythology. From this background, he posits that it's even possible the legendary Palamedes really did invent the (Greek) alphabet.
The Greek alphabet originally had only 5 vowels; the additional, long ones were added over time.