This isn't what you're asking about really, but you might be interested to know that petroleum (in the form of bitumen or asphalt) actually has a very long history in the Middle East. There were areas in Iraq, western Persia, and parts of the Arabian Peninsula where bitumen naturally pooled at the surface, and it was found to be useful for waterproofing houses and boats -as well as for construction, miscellaneous sealing, and later incendiary military applications- as early as the Ubaid period in Mesopotamia, from the fourth millennium BCE onwards. These simple, pitch-covered reed boats were sea-worthy enough to allow the Ubaid people to sail along the whole of the Persian gulf, down to the land called Dilmun (identified with modern Bahrain). It's possible that the trading center of Dilmun utilized bitumen from Arabia (although the substance was so copious in Mesopotamia, in multiple forms including bituminous rocks and natural oil springs, that there was surely no strong incentive to import it).
As for modern Saudi Arabia, unfortunately I haven't had an easy time finding much information on the development of oil extraction as a pivotal strategic resource in the 20th century. I'll edit this post if I turn up any good info throughout the night.
The critical figure in this, as in so much of the 20th century history, is Winston Churchill. In 1909 the first British naval ships powered by oil rather than coal were launched, but it was Churchill as First Lord of the Admiralty from 1911 onwards who committed to the full-scale conversion of the British fleet, including its capital ships, to run by oil and did so at the height of the Anglo-German naval race.
However, while Britain at that time had significant amounts of coal, in the early 20th century it had no viable oil reserves (this would have to wait until north sea oil). As a result, the very strategic survival of the British empire, which had always depended on naval superiority, now also required a massive (for its time) and strategically secure quantity of oil that could only come from overseas.
Now, it's worth noting here where the oil fields in this region are actually located. https://en.wikipedia.org/wiki/Oil_reserves_in_Saudi_Arabia#/media/File:Regiones_petroleras_-_Medio_Oriente.svg
You can see from this map that it's not really the Arabian peninsula as a whole, but really areas bordering the Persian Gulf that are where virtually all of that oil is located. From 1820 (long before oil was discovered) Britain signed signed the first of its treaties with the Gulf countries to guarantee British shipping and trade to and from India and as a result the Persian gulf became basically a British lake and the surrounding Sheikhdoms entered the British sphere of influence if not becoming actual protectorates. As a result, even before oil, the Persian Gulf was a notable, albeit not critical part of Britain's trade empire.
With those political and imperial relationships already well established, the first of the Persian Gulf region's oil deposits was discovered in what is now modern Iran by the Anglo-Persian Oil Company in 1908, just as the British navy was starting to convert to oil, giving Britain that critical exclusive access to an oil reserve on friendly terms that was desperately needed to supply its navy. The vital strategic nature of oil and its rapidly increasing price led Churchill to pursue a policy of a wholly government owned and controlled supply of oil. Further to this, and given that controlling Persian oil was the best way to pursue this, the only deep-water port in that part of the Gulf at this time was the port of Basra, which meant that by the time World War I broke out in 1914, and as a result of British Navy contracts with the by-now government controlled Anglo Persian Oil Company, the entirety of the navy's oil supplies were coming from Persia and through Basra, a port in the hands of the enemy Ottoman empire. As a result Britain invaded Ottoman Iraq to secure Basra in November 1914 just two months after Britain entered the war.
By this point, formal exploration was far from complete, it was widely assumed that the Persian Gulf region would contain vast quantities of oil beyond what had been discovered in Persia. Britain bought the exploration rights to Kuwait in 1913 and for Bahrain in 1914. By the end of the war it was widely understood that the northern Iraqi province of Mosul was likely to have large quantities of oil which became a major sticking point on how that country should be formed even though commercial extraction didn't start til the late 1920s.
Despite the British grab for oil concessions and favorable regional position, American oil companies secured some of the best deals and made some of the most significant discoveries. This was a combination of the comparatively poor British relationship with the Saudis and the false British assumption that Saudi Arabia wouldn't have any oil. In 1938 SoCal struck oil Eastern Province after earlier large American discoveries in Bahrain and Kuwait.
Unfortunately my knowledge of the exploration/commercial exploitation of other parts of the Persian Gulf region starts to drop off from about 1940 until certain aspects of the history of the regional oil industry from about the late 70's onwards. Resource nationalism from the late 1950's was spurred by the realization that oil profits, which dominated the sovereign budgets of these countries, was being determined by the prices set by western oil corporations. The inherent imbalance of this (a corporation having basically existential power over it's "partner", an actual sovereign nation) contributed to resource nationalism and the founding of OPEC in 1960.
In terms of infrastructure, the middle of the 20th century saw significant investment from each of these countries, particularly in contrast to the original situation where in 1914 Basra was the only viable oil port. The Persian Gulf states have each invested over the course of the 20th century in the creation of an independent port capability, and that investment in infrastructure continues to this day.
One last thing that's worth noting is that so far I've discussed oil, but not all of the well known "Gulf oil states" actually have all that much crude oil. Qatar might well be considered alongside the UAE and Saudi Arabia as another wealthy Gulf oil state but their wealth and reserves are almost entirely in natural gas which is why they were not nearly as wealthy and were far less prominent for most of the 20th century until, in the 1990s the extraction and processing technology became sophisticated enough for Qatar to become one of the world's leading exporter of liquefied natural gas.
Source wise:
Eugene Rogan's The Arabs: A History has a great overview of these oil discoveries and the imperial machinations involved.
You can read a discussion by Churchill on the strategic ramifications of the oil conversion of the British navy from 17 July, 1913 here, from Hansard: http://hansard.millbanksystems.com/commons/1913/jul/17/shipbuilding-repairs-maintenance-etc
Lastly on more modern developments like Qatar, though much of the book covers far more recent events, Steve Coll's Private Empire: Exxon Mobil and American Power is quite good.
Sorry if this repeats CptBuck in part, this wasn't finished by the time they posted.
The first major oil and gas exploration in the middle east didn't happen in the Arabian peninsula, but closeby in Persia. William Knox D'Arcy was an English miner who made his fortune in Australia and New Zealand. He moved to Iran in 1900 with an agreement to prospect for oil in 1901, called the D'Arcy concession, giving him the rights to look for oil throughout Persia. It took a few years, but they struck a sizeable amount of oil in 1908 and formed the Anglo-Iranian Oil Company (AIOC), which would in 1954, become British Petroleum, otherwise know as BP, still around as a public limited company. Riley 'Ace of spies' claimed to have been involved but it's only from one likely fictional account.
It's worth remembering at this time oil and gas (in which Persia is a bit richer), weren't yet very major products or industries, except perhaps in the US. Gas and whale oil lighting and was being replaced by electric, while coal still dominated home heating with coal chutes and for boilers, notably on Dreadnoughts. Cars were still in their infancy and mass production only began in 1908 in America (using domestically produced oil, of which there where large easy early finds of 'rock oil' in Pennsylvania, years before), in Europe in France not until after the war. Also the reason a Brit was in the region was that Persia was at the cross roads of the great game of three empires in the 19th century, at stake were economic spheres of influence, buffer states and Russia always wants warm water ports.
The French and British had an interests (maybe from geological studies) in the oil and gas resources of Mesopotamia, inconveniently Ottoman. Coincidentally a fair bit of archaeology occurred in the period (both T.E. Lawrence and Schliemann were archaeologists among many others). They tried via the Iraq Petroleum Company (IPC), which in 1912 became the Turkish Petroleum Company (TPC), formed with the purpose of acquiring concessions from the Ottoman Empire to explore for oil in Mesopotamia. The owners were a group of large European companies, the largest single shareholder was the British Government-controlled Anglo-Persian Oil Company, which by 1914 held 50% of the shares. TPC received a promise of a concession from the Ottoman government but the outbreak of World War I in 1914 put a stop to all exploration plans.
The Germans had influence with the Ottomans, and were keen on expanding their dominance of Eastern European trade by building a railway from Berlin to Baghdad (and block Russia aspirations in the Balkans), with the intention of possibly extending it all the way to Basra on the Persian Gulf. This would have obviated the necessity for as large a naval presence (both military and commercial) to compete with British interests in the region. It would have the added benefit of avoiding the lengthy shipping routes and tariffs, as well as a port for exports. Meanwhile the Saudi peninsula was still a largely undeveloped backwater, also Ottoman, much like what you might have seen in a David Lean film.
Anyway, with this in the background. The Royal Navy (under Churchill), saw the advantage in moving from coal to oil, around 1914. Some claim the threat of the completion of the Berlin-Baghdad Railway (Berlin to Constantinople is part of the 'orient express') was an aggravating factor (largely kept hidden from the public) helping incite the onset of hostilities in the Great War. The first regiment (2nd Battalion) to be deployed in the war, was the Dorset Regiment in the Mesopotamian campaign (Mesopotamia Expeditionary Force / Indian Expeditionary Force "D"). Their orders to protect the oil refineries, tanks and pipeline at Abadan and cover the landing of reinforcements if these should be required. Only if hostilities with Turkey were to become fact should they try to occupy Basra too, which they did in 1914.
Anyway you get the war and then in 1916 Sykes-Picot, carving up Mesopotamia between British and French interests in their mandates. After the war, they managed to exclude US interests (probably the then global leaders in production and demand) via the San Remo Petroleum agreement in 1920, so the Americans started to look elsewhere close by, Bahrain. The period between the wars is covered well here. Standard Oil of California (SoCal) through its subsidiary Bahrain Petroleum Co. (BAPCO) struck oil there in 1932, close to the Saudi border. In 1933 the very new neighboring Saudi Arabia granted a concession to SoCal to explore, who assigned it to California-Arabian Standard Oil Company (CASOC). They did poorly initially, but in 1938 they struck a gusher in Dhahran, and the rest as they say... In 1944 company name was changed from California-Arabian Standard Oil Co. to Arabian American Oil Co. (or Aramco). In 1951, the company discovered the Safaniya Oil Field, the world's largest offshore field, and in 1957, the discovery of smaller connected oil fields confirmed the Ghawar Field as the world's largest onshore field, both generating most of the prosperity of Saudi Arabia. In November 1988, a royal decree changed its name from Arabian American Oil Co. to Saudi Arabian Oil Co. (or Saudi Aramco, by which it's know today) and took the management and operations control of Saudi Arabia's oil and gas fields from Aramco and its partners. It was the world's largest company in terms of estimated market value in 2005. *grammar, spelling and removal of links.
