What group of people moved into Italy from Asia Minor and took control of Rome around 600 BC, bringing with them their building techniques and religious beliefs?
The ancestry of the Latin peoples and their immediate neighbours is not well known, there is little documentation to go on, however, none of them seem to have come via Asia Minor.
Early Rome's history is shrouded in myth and wrapped in legends -- many of them manufactured centuries later. What facts are known largely come from Archeology. The Bronze-Age settlers of central Italy (the Latins) had growing populations and moved down from the more defensible mountain valleys into the more fertile coastal region and founded a number of fortified villages. One seeming cluster of villages in the middle of the 8th Century (the 750s) became Rome.
There seems to have been an earlier settlement on the Capitol since the 1400s. There was also a farming village on the Palantine since the mid-900s. The origin of Latins and most other early Iron Age tribes in Italy is almost unknown, but it seems they migrated down from what is now Hungary/Austria sometime in the volkerwanderungs of the 12-13th Centuries BC.
The Etruscans are equally mysterious, although modern genetic studies suggest they were an offshoot of whatever people it was that invaded the Eastern Mediterranean during the same volkerwanderung. We really do know much about that event and what caused it. However, they were literate by the 8th Century and well established.
As Rome gathered power and wealth, a lot of Romans -- as ancients were wont to do -- tried to graft their own history onto older legends. The whole idea that the Romans were somehow descended from the Trojans appears in late 2nd Century BC and gathered strength, particularly a hundred years later as Caesar Augustus needed more 'antiquity' than the average Roman.
The old Latin Capital of Alba Longa had supplied many of the old Patrician families (including the Julii) when Rome absorbed it; so the Trojan myth was assigned to create even greater antiquity and more prestigous roots.