Establishing Targaryen rule
Aegon I Targaryen, "Aegon the Conqueror", was crowned King by the High Septon when he entered Oldtown in the year 2 AL.
The naming of the "After Aegon's Landing" (or "AL") dating system is inherently a misnomer, as Aegon I himself counted the years of his reign as starting from the end of his conquest, when he entered Oldtown and was blessed by the High Septon, which occurred two years after Aegon and his army first landed on the mainland at the mouth of the Blackwater Rush. It would have been more accurate to call it "AC" for "After Conquest". Aegon originally declared himself king when he first landed at the mouth of the Blackwater, but in later years he counted the years of his reign as beginning with his crowning by the High Septon. Perhaps this was meant to reinforce a sense of his legitimacy.
There is little information about the rest of Aegon I's reign except that he ultimately died in 37 AL. Apparently, he ruled efficiently and well for almost forty years, working to unite the Kingdoms. Aegon's unification finally stopped the endless petty wars between the Seven Kingdoms over minor glories, territory, and riches. In time, the peace brought by the new Targaryen order made Westeros prosper as it never had before.
Aegon had already assembled the first Small Council during the Conquest, which now set about organizing his realm. He named his bastard half-brother Orys Baratheon as the first Hand of the King, a position he excelled in for decades. Having overthrown House Durrendon in the War of Conquest, Orys founded House Baratheon by marrying the daughter of the last Storm King, taking his sigil, words, and seat at Storm's End. House Baratheon continued to rule the Stormlands under their Targaryen cousins.
Construction of King's Landing
Probably the biggest "domestic" policy project that Aegon launched (apart from ending the constant wars between the Seven Kingdoms) was to found an entirely new capital city for his unified realm, at the site of his first landing on the continent at the mouth of the Blackwater. The scale of the project was staggering: on a scale of decades, the Targaryens intended to construct a new capital city bigger than Oldtown itself, which had taken millennia to reach its current population.
Aegon I first fortified his initial landing point at the northern shore of the mouth of the Blackwater River, a strategically very defensible position protected to the east and south by water, and surrounded by open plains which would expose any attacking army long before it arrived. After the conquest, Aegon ruled for some time at the crude boomtown which sprang up there, known as "Aegonfort". However, it remained a ramshackle pile of wood and earth, so after some time he moved his seat back to Dragonstone, until such time as a proper new capital city could be built in the former location of Aegonfort. He had Aegonfort razed, while planning and construction of the new capital city of King's Landing began.
By the time Aegon I died, however, only the foundations of the city and tunnels of the future Red Keep were finished. Aegon continued to rule from Dragonstone for the rest of his life, though he always intended to some day move the royal seat back to Aegonfort/King's Landing. The city was only finished during the reign of Aegon's second son, Maegor the Cruel. The Iron Throne itself, however, was not taken to Dragonstone, because it was too big to move.
- Keep in mind that the Iron Throne in the books is a massive 30 foot tall construct literally made out of thousands of swords. George R.R. Martin has stated that he does like the TV version's design, but it could never hope to be built on the vast scale in his books, in which the interior of the Red Keep is a on the scale of St. Peter's Basilica in Rome. Martin does sympathize that the TV series already films in the largest studio in Europe, Belfast's "Paint Hall", and there was simply no way to build it on this scale (thus Littlefinger's scene in "The Climb" where he mocks legendary historical figures by counting that the Iron Throne doesn't literally have "thousands" of swords in it is accurate for the TV version - but the book version of the Iron Throne actually is made of thousands of swords, and so tall that the king has to ascend a 30 foot flight of steps to reach the top.
King's Landing was rapidly constructed within the first century of rule by the Targaryen dynasty. This rapid expansion didn't keep pace with organized "city-planning", resulting in numerous slums forming such as the district known as Flea Bottom. Many of these slum districts are poorly organized, ugly, and filthy because their placement wasn't thought out during the rush to slap down new buildings. In contrast, the second largest city in Westeros, Oldtown in the Reach, is thousands of years old, but grew only in incremental steps which were well planned out beforehand, resulting in it being a much better organized and beautiful city without the squalid slums of King's Landing, while being only slightly smaller than the capital itself.
Aegon also pursued an alliance with the Faith of the Seven, recognizing that he could not easily rule over an entire continent's population if their religious leaders opposed him. Even Aegon tread lightly with the leaders of the Faith, winning them over with the carrot instead of the stick. He knew that should the Faith oppose him, it wasn't a simple matter of killing the lords who led a local rebellion, as they were a widespread organization which could stir up broad opposition to the Targaryens if they felt cornered.
Aegon even converted to the Faith of the Seven just prior to the Conquest, and the rest of House Targaryen followed suit, though this was probably a calculated political action meant to smooth his acceptance in the Seven Kingdoms. Crucially, however, Aegon did not cease the old Valyrian customs of incestuously marrying brother to sister in order to "keep the bloodlines pure". Moreover, Aegon also continued the Valyrian practice of polygamous marriage, thus he remained married to both of his own sisters, Rhaenys and Visenya. It is unknown if subsequent Targaryen generations ever practiced polygamy, though apparently that practice did not last beyond Aegon's generation. Either way, the lords of Westeros weren't quite as upset with that as they were incest (given that many lords had mistresses were were wives in all but name, and even an official naming system for noble-born bastards). Nonetheless, all of Aegon's children were incestuously born to his own sisters, which caused increasing tensions with the Faith of the Seven, whose moral teachings expressly forbade such an incredibly high degree of incest. The Targaryens considers themselves dragons among sheep, akin to gods looking out over lesser men, and thus above the rules and social restrictions of commoners. Aegon himself, at least, had the power to back up such grandiose claims and cow the Faith into silence, but Targaryen incest practices would lead to major problems for his sons after his death.
Tensions with Dorne: The deaths of Rhaenys and Meraxes
A key point is that Aegon formally declared himself king of all of the "Seven Kingdoms" when he first landed in Westeros, but the Targaryens never actually managed to conquer Dorne. Technically the title he took was actually "King of the Andals, the Rhoynar, and the First Men" as well as "Lord of the Seven Kingdoms". Despite the fact that the Targaryens didn't actually control the still defiantly independent Dorne, however, they still continued to legally claim this full title. Basically, when centuries later the North rebelled against King Joffrey and declared themselves independent, Joffrey didn't instantly start calling himself lord of only six kingdoms: he simply considered Robb Stark's rule of the North to be an unlawful rebellion. Taken to an extreme, despite the fact that the Targaryens never conquered or controlled Dorne, they still maintained the claim and considered themselves kings in absentia. Thus Aegon I was still formally called "King of the Andals, and the Rhoynar, and the First Men" until the day he died, as did all of his successors.
Little is known of the condition of independent Dorne during the two centuries that it existed beside the unified Targaryen realm, but it was presumably in a precarious state and walked a political tightrope. It's unclear if Aegon I ever formally made peace with Dorne or if they were still technically at war for decades. Queen Rhaenys had failed in her initial invasion of Dorne during the main Conquest, but it is possible that multiple renewed expeditions were made against Dorne during this time.
It is stated that ultimately, Queen Rhaenys and her dragon Meraxes "died in Dorne", though exactly how has not yet been mentioned. What is certain is that Meraxes was killed through sustained crossbow-fire which pierced its eyes: the Dornish were actually able to kill a dragon. This wass never treated as an impossibility: even Aegon himself was wary of committing all three dragons at the Field of Fire, because he recognized that while dragons are formidable they are not outright invincible, and may be eventually killed by sustained arrow fire. The Dornish somehow ambushed Rhaenys on Meraxes with many companies of crossbowmen.
Queen Rhaenys was survived by her children with Aegon I: their son Aenys and two or three daughters. The initial three Targaryen dragons produced subsequent generations of progeny, so Meraxes may have left some eggs behind (which might have even hatched by the time she died) - though little information has been given so far about the descent of the dragons, and the other possibility is that all of the later dragons were the progeny of only Balerion and Vhagar.
The mind boggles that Aegon did not pursue vengeance against Dorne, but his resources may have been stretched to the limit as it was holding his new realm together. Moreover, Rhaenys was always Aegon's favorite sister-wife, and the loss of her may have broken part of his spirit to fight. Whatever the case, Dorne remained "Unbowed, Unbent, Unbroken", while Aegon withdrew back to Dragonstone as construction on King's Landing continued.
Aegon eventually died on Dragonstone in 37 AL, and was cremated according to Valyrian funeral custom.