We've had a number of good posts on the migrations recently. While I type up a more specific answer, please have a look at Alriclofgar's answer on the Saxon migration here (this one about the role of religion in antiquity might also interest you). Also have a look at this megathread we did some time ago about migration in the past. Here is Shlin28 talking about when the Roman empire ended and barbarian kingdoms emerged. Bitparity talkes about Roman cultural identity here. Here we have GeorgiusFlorentinus talking about Visigothic identity (and how it was very different from modern concepts of ethnicity). Thejukeboxhero talks about the relation between Roman local aristocracy and the Franks here (but also have a look through his excellent posts on religious changes in the migration period).

Standard works for the Migration period are Heather, Empires and Barbarians, Wickham, The Inheritance of Rome and Framing the Early Middle Ages, and Halsall, Barbarian Migrations and the Roman West.

I'm going to recommend People and Identity in Ostrogothic Italy by Patrick Amory. It should give you at least some picture of Germanic ethnic identities existing alongside Roman ones.

Your actual question is incredibly vague. What was an immigration like? Any answer would have to be very very broad, and writers at the time didn't worry so much about the details of cultural assimilation. Today, we understand that ethnic identity is fluid, and that means we have peoples who are in some nebulous state between Germanic tribe and romanized polity that is still distinct from the empire.

The Vandals, Suebi, and Alans crossed the Rhine in 406. They settled in Hispania, briefly. The Suebi stayed, while the Vandals and Alans invaded North Africa. None of these migrations and settlements were sanctioned by Rome. We have records of Vandalic still being used in North Africa, and its unclear how assimilated the Alans (an Iranian speaking people) were with the Vandals. After Belisarius defeated them, they were dispersed and ceased to exist as a cohesive ethnic group. Likely, Vandal and Alan identity would not have lasted more than a generation or two after that. The Suebi remained in Hispania and were eventually absorbed by the Visigothic kingdom. After a certain point, it becomes difficult to tell whether certain historical figures with Germanic names from the region were Suebi or Visigothic, or if anyone even cared.

The Ostrogoths were sent to take Italy from Odoacer. Theodoric the Great ruled ostensibly as a viceroy from the Eastern Roman Empire, though today we acknowledge that he was effectively independent and talk about his kingdom as a separate state. We know that the Ostrogothic army contained Huns and Rugians, and the current trend in academia is to make a case for them being multiethnic, but I haven't found any reason to believe non-Germanic presence among them was terribly significant. We have land deeds from Italy in Gothic, and it seems likely that the language was still in use in Theodoric's Kingdom, though most Ostrogoths would have likely spoken Latin also. Goths and Romans were treated separately by law, with Goths serving in the military and Romans filling administrative roles. Both farmed. We know that they intermarried, but this doesn't seem to be the norm. After Justinian defeated the Ostrogoths, they were dispersed like the Vandals. Some went off to join the Franks, some took up service in Justinian's armies, and some settled in Northern Italy and the Balkans, integrating with the local populations.

The Visigoths and Franks both served Rome as foederati. This means that their armies served as armies for Rome in exchange for land to settle and farm. The Visigoths themselves had a hand in creating the need for foederati, having destroyed the Roman army at the Battle of Adrianople. Their presence within the borders of the empire was sometimes welcome and sometimes not. By the time the Visigoths were given the land that would become their successor state, most of them had spent much of their lives within the empire. They would have spoken Latin and been fairly acclimated to contemporary Roman culture. Whether or not the Visigoths even spoke Gothic at all at this point is something hotly debated by gothicists. They became the ruling class in Visigothic Spain, but over time the locals began adopting Gothic names and the ethnic distinction between Goths and Romans began to blur, becoming more of a legal/class distinction. Some of the distinction between Goth and Frank also began to blur in border regions. The Visigoths absorbed the Suebi, converted to Trinitarianism, and were defeated by the muslims. Visigothic idenity remained for quite a while afterwards, with muslim sources referring to locals as goths, and many noble families claiming Gothic descent.

Religion was perhaps one of the largest cultural differences between migrating Germanic peoples and Romans, particularly when many of these Germanic peoples were already Latin speakers. Romans were usually Trinitarian Christians. Most of the Germanic groups we see entering the empire had already been converted to Arian Christianity. They maintained separate churches and a separate clergy. The Franks are notable for being the only Germanic people we see converting to Trinitarianism right away.

The Anglo-Saxon invasion is a whole other matter entirely. Bede would have us believe the invaders massacred all the locals and took their land, but genetic studies (see Brian Sykes) show that this isn't the case. It's more likely that they killed or drove off the local Brythonic nobility and the locals assimilated into their culture. King lists show early Anglo-Saxon kings with Brythonic names, and one of the first things ever written down in English was by a man with a Brythonic name, Caedmon. It's interesting to compare the Anglo-Saxons with continental Germanic peoples, because in England we see the ruling class assimilating the locals, but in Hispania and Frankia we see the locals assimilating the ruling class.

Barbarian Migrations and the Roman West by Guy Halsall has a fantastic, in-depth discussion of these issues.

In short, it depended on whether there was a vacuum in power, or residual power structures from the Roman Empire remained... In France, for example, the Goths and Franks adapted more to the residual Roman culture in order to reinforce their rule... you can observe this in the French language, which is closely related to the Vulgar Latin which was spoken at the time. On the other hand, in Britain there was a real power vacuum, so the local people adopted a more barbarian identity, and a Germanic language became dominant.