PS: if the button does not work, please tell me via commenting on one of the fics in this series (feel free to comment if you’re interested in more details as well, I’d love to talk your ear off), thanks :) Because of AO3's character limitations, I had to condense details.
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Visenya and Aegon are both alphas and thus never got together. Aegon married Rhaenys, who is an omega; Rhaenys had two sons: Aenys (omega) and Maegor (alpha). Because Aenys is an omega, Maegor becomes heir and later king.
Aenys is first married to Alyssa Velaryon (alpha), though all three of their children carry the Targaryen name: Rhaena (alpha), Aegon (alpha) and Viserys (omega).
King Maegor I later elopes with Aenys, and they also have three children: Jaehaerys (alpha), Alysanne (omega) and Vaella, who Aenys dies giving birth to. Vaella doesn't survive the night.
After Aenys' death and Maegor I's subsequent insanity, Rhaena, the eldest alpha of Aenys' children, rebels against her uncle/stepdad Maegor I. She wins, marries Alysanne and is named Rhaena I.
Rhaenyra is an alpha, so is Aegon (II). The Dance of Dragons still happen, Aegon briefly rules as Aegon II before Rhaenyra becomes Rhaenyra I.
Jacaerys (alpha), Lucerys (omega) and Joffrey (alpha) are Laenor's (omega) bastard sons. Daemon (alpha) claims them as his and when they marry, Viserys I legitimizes them as Daemon's.
Rhaenyra (alpha) marries Laena (omega), their children are as follows: Baela (omega), Rhaena (omega), Aegon (III, alpha), Viserys (omega), Visenya (stillborn).
Jacaerys is arranged to be married to Baela; Lucerys marries Aemond (alpha), their children are: Jaehaera (alpha), Jaehaerys (omega) and Maelor (alpha); Joffrey marries Daeron, omega son of Viserys I and Alicent Hightower.
Aegon III marries Jaehaerys, their children are: Daeron (omega, Daemon I Blackfyre's mother), Baelor I (alpha), Daena I (alpha), Rhaena (beta) and Elaena (omega).
Viserys marries Maelor, their children are: Aegon IV (alpha), Aemon (omega) and Naerys (alpha). Aemon and Naerys switch roles; Aemon is Aegon IV's queen, Naerys is the Dragonknight.
Aegon IV era happens very similarly to canon, except Daeron is Daemon I Blackfyre's mother and Aemon is Daeron (II in canon, not to be confused with the other Daeron)'s mother.
Daeron II's (Daeron, first of his name, in this fic; Daeron I and his sister Daena switched roles, Daena was king) kids are as follows: Baelor (alpha), Aerys (alpha), Rhaegel (omega), Maekar (omega).
Baelor Breakspear is originally married to Jena Dondarrion, and together they have two alpha sons: Valarr and Matarys.
Aerys is childless as in canon.
Rhaegel is married to Daemon I Blackfyre. They have several children, the most notable being Daemon (omega), Haegon (alpha), Aenys (omega), Calla (alpha).
Maekar and his elder brother Baelor eventually elope, much to their parents' chagrin. They have six children: Daeron (alpha), Aerion (omega), Aemon (beta), Daella (beta), Aegon (alpha) and Rhae (beta). Aerion is betrothed to Valarr.
The First Blackfyre Rebellion still happened, but with the exception of Daemon (II), Aegor Bittersteel manages to escape with Daemon I's remaining children. Rhaegel and Daemon (II) are both placed in house arrest.
The Trial of Seven never happens. Valarr, Matarys and Daeron II all die during the Great Spring Sicknes in 210; Baelor Breakspear succeeds his father as Baelor II.