This is a big question, but I can hit the high points of the political and military history. Culturally, the Etruscans lived on much longer than their actual cities and political influence. As the Romans defeated individual Etruscan cities, the routinely "stole" their gods via evocatio. The city of Rome was a part of a widespread homogenous culture in northern and central Italia in the 6th and 5th centuries BCE, and much of the cultural identity of this diverse region owed its origins to Etruscan cities. Linguistically, there were sharp divisions, but as far as art, religious belief, etc, there was a common cultural identity from the Po all the way to Campania. As the Etruscan cities were defeated by expanding Roman power, these cultural markers did not disappear, not in the conquered cities and certainly not in Rome. I will leave the specifics to someone else.

The most famous and perhaps the earliest clash between the Romans and the Etruscans proper was the Roman siege of nearby Veii in 396 BCE. Livy's description sounds a little too much like the siege of Troy, but there you have it. It resulted in Rome annexing the Etruscan city's territory, which was enormous compared to what Rome nominally controlled up to that point. A lot of historians call this the first major expansion of Roman power beyond the ultra-local bounds of the seven hills and the marshy spots by the Tiber river. The conquest of Veii was followed almost immediately by the violent visit of the Gallic tribes to Rome, a trauma which the Romans would remember for centuries to come. The VIPs of the Romans, including the Vestal Virgins, were sheltered by the kind denizens of Etruscan Caere map. Rome repaid their kindness by offering them a series of Roman civic rights, ultimately resulting in the city being granted Roman citizenship (but without the right to vote in Roman elections). At around the same time, the Romans planted two citizen colonies in the "gates of Etruria" (Livy 6.9.4) north of Rome. So there you have three different early methods of Roman expansion and extension of state power: outright conquest; political absorption via granting civic rights; and occupation of territory via colonial foundations.

There was no centralized Etruscan power--no Etruscan High King; no Etruscan federal council; no Etruscan senate. There were numerous Etruscan city-states, and they sometimes worked together for common goals, sometimes fought with each other, and usually did their own thing. This was the sort of situation in most of Italy in this time period, and is one of the reasons Rome was able to whittle away from the center so effectively. After Veii and the planting of the coloniae, twelve of the Etruscan cities (we don't know which) formed a collaborative League at their sacred site Fanum Voltumnae. We don't really know how it worked or what exactly they did, but in 358 (so decades later) the powerful city of Tarquinia declared formal war on Rome, with Caere (probably) in tow. The conflict resulted in a truce, with no territories changing hands, perhaps because Rome was busy with the dangerous Samnites in the South.

In 311, one of the colonies (Sutri) was attacked, but the Romans recovered it quickly and then seized territory along the Tiber north (perhaps as far as Volsinii (map). This push included Roman (separate) treaties with Cortona, Arretium, and Perusia to the Northeast. These treaties, in very basic terms, served to divide the cities and prevent collective action, and also ensured the dynamic remained between Rome and individual cities, not Rome and any League.

However, after the northern Tiber push, the Etruscans got organized and formed a coalition with the Samnites and the Gauls. Their powerful combined force was defeated by the Romans at Sentinum in Umbria in 295 BCE. Many historians consider this the last collective action of any multi-city federalized "Etruscan" group. After this battle, the Romans reduced the other cities one by one, including Volsinii (now on the frontier of Roman expansion) and Vulci. Caere was among the last, having finally defied Rome power in 273. Her territory was confiscated and redistributed in a series of Roman maritime colonies. Falerii and Volsinii were eventually destroyed completely after uprisings, during which process the sacred site of Fanum Voltumnae was looted and destroyed. The Roman general Flaccus supposedly carried back dozens of bronze statues from the site (Pliny NH 34.16.34).