Proto-Sinaitic, also known as Proto-Canaanite, was the first consonantal alphabet. Even a quick and cursory glance at its inventory of signs makes it very apparent of this script's Egyptian origin. Originally it was thought that at round 1700 BCE, Sinai was conquered by Egypt, and the local West-Semitic population were influenced by Egyptian culture and adopted a small number of hieroglyphic signs (about 30) to write their own language. However, recent discoveries in Egypt itself have compounded this scenario. Inscriptions dating to 1900 BCE written in what appears to be Proto-Sinaitic were found in Upper Egypt, and nearby Egyptian texts speak of the presence of Semitic-speaking people living in Egypt.
No matter where and when the adoption of Egyptian signs onto a Semitic language occurred, the process of adoption is quite interesting. Egyptian hieroglyphs already have phonetic signs (in addition to logograms), but the Sinaitic people did not adopt these phonetic signs. Instead, they randomly chose pictorial Egyptian glyphs (like ox-head, house, etc), where each sign stood for a consonant. How did they decide which sign get which consonant? A sign is a picture of an object, and the first consonant of the word for this object becomes the sound the sign represents. In short, this is called the acrophonic principle.
For example, the word for an ox is /'aleph/, which is the first sign on the left Proto-Sinaitic column. It stood for the sound /'/, which is the glottal stop (also written as /?/).
Proto-Sinaitic soon spread to Canaan, hence its other name, Proto-Canaanite, or Old Canaanite script. It evolved locally into the Phoenician script.
Phoenician was the immediately descendent of Proto-Sinaitic. Its major change is the more linear (less curved) shapes of its signs. Other than this cosmetic change, everything else remained pretty much the same. South Arabian was also an early offshoot of Proto-Sinaitic, as its letters are very different in shape and order from Phoenician.
1 The Greek letter that resembles F was called digamma and actually represented the sound /w/. It existed in archaic Greek scripts except the Ionian variant, which supplanted other archaic scripts.
2 The Greek letter that looks like M was the letter san. It appeared in scripts from Corinth and Argos, and represented an alternative to sigma.
3 The letter Q actually existed in Greek for a little while, and it was adopted by the Etruscans before it disappeared due to its extraneous existence.
As you might have noticed, the continuity of the scripts appears quite consistent. For a lengthier treatment on all alphabets, you can go to Alphabet.