Much of the narrative on Gothic history from Wikipedia cannot be trusted. Their 'migration' from Scandinavia is derived only from the sixth-century The Origin and Deeds of the Goths (frequently known as the Getica), written by a man of Gothic origins called Jordanes. Unfortunately, we know very little about the author himself and we need to be cautious about this text. Moreover, this source was supposedly based on a lost Gothic History written earlier by Cassiodorus, a servant of the Ostrogothic king Theoderic. It is impossible to confirm to what extent Jordanes actually used Cassiodorus' text (he says that he was essentially writing from memory after reading the earlier history for three days, then added what he found from his own research), so already the textual history of this source is difficult to ascertain. The Getica did cite other sources, such as Gothic songs and a lost historian named Ablabius, whilst Cassiodorus, as a high-ranking member of the Ostrogothic court, no doubt recorded what some of the Ostrogoths thought about their origins. Nevertheless, nothing here can be relied upon. Origin myths are common throughout ancient and medieval history, such as the Romulus/Remus myth for the Romans, the Trojan origins of the Franks, or the Anglo-Saxon 'invasion' of post-Roman Britain. Their historicity are all questionable, and the same is the case here. We simply have no other evidence that the Goths originated from Scandinavia, so this origin story needs to be treated with extreme caution, if not jettisoned altogether.

So where did the Goths come from? The earliest secure mention of their existence is of their conflicts with Rome in the third century, with the Goths based somewhere north of the Black Sea. Their appearance was not unique, for elsewhere increasingly powerful confederations emerged on Rome's northern frontier - conflict, trade, and cultural osmosis had all transformed the 'barbarian' lands into something rather different and, increasingly, a more threatening place than what the Romans had experienced in the early empire. As succinctly summed up by Michael Kulikowski in his Rome's Gothic Wars (2007):

The answer, at least in my view, is that there is no Gothic history before the third century. The Goths are a product of the Roman frontier, just like the Franks and the Alamanni who appear at the same time. That is clearly demonstrated by contemporary literary evidence, and indeed all the evidence of the fourth and fifth centuries – everything except the sixth-century Jordanes. In the third century, the Roman empire was assaulted from the regions north of the Danube and the Black Sea by large numbers of different barbarian groups, among whom Goths appear for the first time. Not long thereafter, the Goths are clearly the most powerful group in the region, while most of the other barbarian groups with whom they appear in the third century either disappear from the record or are clearly subordinated to them. The most plausible explanation of this evidence is to see one group among the many different barbarians north of the Black Sea establishing its hegemony over the scattered and hitherto disparate population of the region, which was thereafter regularly identified as Gothic by Graeco-Roman observers.

Over time the Goths became dominant over a wider region, to the extent that the Lower Danube eventually became known as the 'Gothic bank' in the 320s, but that did not mean that they were a 'jihadist-like' force of conquest or exceptionally militarily skilled. Mobility and cultural flexibility were (and are) fundamental parts of human nature, so the transformation of identities should not only be seen as the result of conquest. And as /u/iphikrates has so frequently noted (for example here), the Spartans were not super-soldiers either. History, unfortunately, grows with each telling, so we have to be very cautious when we narrate the history of the Goths and be aware of later embellishments. 'Goths' happened to be the name given by Roman historians to a powerful confederation on the frontier, but the ethnic make-up of that group was no doubt far more diverse. Ethnic identity and culture cannot be understood to be the same thing, especially when we essentially have the limited (and highly tendentious) perspective of Roman ethnographers, not of the Goths themselves.

Another group with an equally flexible entry barrier, the Huns, entered the scene in the late fourth century and soon we hear of Gothic groups pleading to be let into the Roman Empire to escape the Hunnic threat. Unfortunately, thanks to Roman mismanagement the Goths revolted and defeated the Romans at Adrianople in 378. By 382 the Romans made peace; as a result the Goths were settled in Roman territories and regularly provided soldiers for the imperial army. This was essentially what happened to many other 'barbarian' groups throughout Roman history, the big difference here being that the Goths were not totally defeated, but instead settled without a comprehensive Roman victory. This speaks of Gothic military prowess, but also of Roman weakness at that particular moment, since Theodosius I, the emperor who had to deal with the mess after Adrianople, could expect no real aid from the Western Roman Empire. Gothic triumph here should therefore be examined from many perspectives, not to be seen only as the result of their innate ability.

Now that the Goths were settled, it's convenient to term them the Visigoths, as it would be useful to distinguish them from another group of Goths who never left their homeland and instead fell under Hunnic domination, the Ostrogoths. Contrary to what you have written here, the Visigoths did not conquer the Western Roman Empire. Instead, their most famous leader, Alaric, ought to be seen as a Roman general who repeatedly tried to find a better settlement with the Roman state and, once that was achieved, the Visigoths in southern Gaul and Spain became a crucial weapon of the empire against other 'barbarians' and usurping generals. I will not go into how the empire eventually 'fell' (I prefer the term 'transformation'), but for more you can take a look at my answer here and /u/iguana_on_a_stick's excellent evaluation of the historiographical debate here. I have also written an answer on the achievements of the Ostrogoths and their downfall here (they started to play an important role in imperial politics when the Hunnic empire collapsed).

Anyway, in this answer I'll instead talk about what happened afterwards to the Visigoths. By the late fifth century, they had been in the empire for a century and their leadership, understandably, was increasingly Romanised. This is why we have Alaric II's legal code, from which we can learn so much about the earlier and quintessentially Roman Theodosian Code, and why the Visigothic monarchy adopted much of the ceremonial aspects of imperial rule (best seen in the increasingly sacralised monarchy from the late sixth century, which bore remarkable similarities to contemporary developments in Constantinople). From my other answers, you will hopefully get an impression that the western 'post-Roman' kingdoms' were all still keen on Roman culture and gave due deference to the emperor in the east, so it should be easy to see how the Visigoths were not behaving any differently when compared to their neighbours.

Lastly, I should address their religion. We know very little about their pre-Christian faith, but we do know that 'Arian' Christian missionaries were active in Gothic territories in the early fourth century, either under Constantine I or Constantius II. Moreover, we also learn that the most famous of these missionaries, Ulfila, was descended from captives from Roman territories taken by the Goths - yet he now had a Gothic name, which speaks volumes for the cultural flexibility I mentioned above. Anyway, 'Arianism' soon became very popular and the Goths would actually be one of the more committed group to hold on to that belief, with the Visigothic monarchy only converting in 587, which is actually a strong indication of the Visigoths' political stability - for all other 'Arian' kingdoms had either converted or had been defeated before then (the extent of the Lombards' 'Arianism' is debatable so I won't address it here).

This is only a very brief summary of the complicated history of the Goths and it is by no means the overwhelming consensus, but I hope it has given you a sense of why Gothic history so complicated and why simplistic accounts found on Wikipedia cannot be trusted without further verification. The same, unfortunately, goes for much of late Roman/early medieval history. We have plenty of sources, but they are incredibly difficult to interpret and none can be taken at face-value. If anyone has any questions, I can try to answer them, but note that I am really not an expert on the 'barbarian' migrations so I can't go into too many specifics :/

Edit: I forgot to cover the end of the Visigothic kingdom in 711! As I can't do better than this answer by /u/thejukeboxhero, I'll just provide a link to it instead :)

[I asked a similar question] (https://www.reddit.com/r/AskHistorians/comments/5vpjx9/migration_period_barbarian_invasions/) and got two good answers that I think is relevant to your question.