Latium (or Latinum): Part of the journey of Aeneas takes him to the underworld where he sees his father, who has died. Anchises advises Aeneas to settle in Latium above the Tiber River (but not yet in Rome.) Aeneas goes to Latium, signs a treaty with the local king (Latinus) and marries the king's daughter, Lavinia, naming his new town after her. This leads to a battle with her former boyfriend, Turnus, the king of the Rutulians, and both Turnus and Latinus are killed. Aeneas then creates the new Latin state from the survivors of both sides. Time passes.
Alba Longa: Iulus Ascanius founds a new city at Alba Longa (now Castel Gandolfo) and his son Silvius (Aeneas' grandson) founds the Silvian dynasty, which lasts for approximately 300 years. In the eighth century BC, Prince Amulius of the Silvian dynasty, deposes his older brother Numitor and kills Numitor's male children. Numitor's daughter, Rhea Silvia, is forced to become a Vestal Virgin. Numitor survives the coup, but is a prisoner.
Depending on which version you believe, Rhea Silvia either prays to Mars or he just shows up on her doorstep, but the liaison results in the birth of Romulus and Remus. Amulius finds out about the birth of the twins and orders them to be slain.
Rome: The servants, instead, either hide them in a basket on the riverbank or just put them down in the rushes near the river, where Rome now stands. A she wolf finds them, suckles them, and takes them to her cave. A shepherd, Faustulus, later finds them there and raises them as his sons.
The twins reach maturity and discover the treachery of Amulius. They kill Amulius, free their father and restore him to the throne in Alba Longa, and decide to found their own new city on either of two hills on the bank of the Tiber, where they had grown up. They use augury to determine who should be king, but the results are ambiguous: first Remus sees six vultures from the Aventine hill, then Romulus sees twelve from the Palatine. Remus claims he should be king because his vultures appeared first. Romulus says he has more vultures, so he should be king. There are several versions of what happens next, but they all finish with the death of Remus and with Romulus as the sole ruler. He becomes the first of Rome's seven kings with his city on the Palatine.
The new kingdom needs more people, so Romulus invites in local outcasts and bandits. They are short of women so they trick a local Latin tribe, the Sabines, into coming to the forum for a peace conference, which is celebrated by games and carousing. When the Sabine men get drunk, the Roman men seize and carry off their virgin daughters and quickly rape them all so that nobody else would want them. The daughters, being women of their time, love this kind of treatment, and send messages to whichever of their relatives have survived the party, saying that they do not want to be rescued from the virile Romans. Peace is eventually restored, but the Roman kingdom starts its expansion.
Behind the myths: There is no evidence that Aeneas ever reached the shores of Latium (or that he even existed, for that matter) and this whole part of the story may well have been concocted to give the Romans a noble history. They can not demonstrate descent from the Greeks, so why not be Trojans?
Romulus and Remus may well have been from Alba Longa, a town established by a Latin tribe: there are cultural affinities. But another theory is that they were just Etruscan outlaws encamped near the Tiber at the place where Etruscan and Latin tribal territories met, and only twins in the sense that they had similar backgrounds and goals and in that they each had a group of followers -- they were evenly matched and alike as twins.
The abduction and rape of the (Latin) Sabine women explains how the agglomeration of multiethnic outlaws on the Capitoline hill becomes part of the Latin language group. The Latin Sabine women raise their kids in the Latin culture.
The fight for primacy between Romulus and Remus, presaged by the Amulius/Numitor struggle, and predicting the Roman civil wars, is usually interpreted as a mythic telling of how the Palatine settlement, which became Rome, achieved primacy over the other pre-Roman settlement on the Aventine. (The ancient Romans had found traces of both pre-Roman settlements and needed to explain why the city went up where it did.)
One version of the myth of how Remus was killed says that the fight with his brother began when Remus mockingly jumped back and forth across the sacred city boundary (the Pomerium) that Romulus had laid out on the Palatine. The story of the death of Remus for his act of impiety was used by the Senate to reinforce the rule that military leaders were not allowed to bring their armed forces into the city of Rome (enclosed by a Pomerium) or into any other non-military Roman city unless they were invited to enter as part of a Senate-approved triumphal march along the established route. This version of the myth was very popular with the Senate and later with the Pompey faction, when Julius Caesar was coming toward the city after crossing the Rubicon -- even though Pompey had earlier used his own troops to "restore order" in the city. The Liberatori also promoted it after they killed Julius Caesar on the Ides of March.
Julius Caesar, of course, claimed to be a descendant of that Iulus Ascanius, the son of Aeneas, who founded Alba Longa. The famous Brutus, who was the figurehead of the Liberatori plot that killed Julius Caesar, claimed to be a descendant of the Brutus who killed the last of the seven Roman kings.
Augury, strictly speaking, is only that branch of magic that interprets the flight or actions of birds. The "Augur" branch of the Roman priesthood later also used other methods to determine proper courses of action, much to the dismay of the other priestly specialists. The word may originally have been something like avgury or avigury (there was no real letter U in the Roman alphabet) from the Latin word avis, which meant bird.