It is widely understood among historians that the Goths originated from Sweden, but this particular question has been scrambling around in my head for quite a while and I haven't found any conclusive research on it. Maybe I'm just bad at finding it.
Considering the presence of Goths in so many names of peoples and geographical areas and provinces in Sweden (Götar/Geats/Gautar/Gutar/Goter, Göteborg/Gothenburg, Götaland, etc) it almost feels obvious that these are all historically linked, however if history has taught me anything it's not to be superficially presumptive.
Considering that Visigoths literally means "West Goths" and the province of Västergötland literally means "West Gothland" (lat: Westrogothia), just as Ostrogoths literally means "Eastern Goths" and the province of Östergötland literally means "East Gothland", the connection almost seems too obvious. Like I almost can't believe it could be that simple.
Were they already two distinct subgroups of Goths that set out to migrate together and who later split up into their original groups again, or were they all the same people until they split up during their migration?
Can someone shed some light on this for me?
It is widely understood among historians that the Goths originated from Sweden
I would not say that this is the case. Peter Heather, for one, argues that the Goths more likely originated (coalesced, perhaps?) on the Baltic coast of Poland.
In any case, the division of the Goths into Ostrogoths and Visigoths occurred long after the Goths left the Baltic and settled in the modern Ukraine. Probably due to pressure from the migrating Huns, many Goths sought refuge within the bounds of the Roman Empire; not all were admitted, and some, presumably, didn't even try. The Ostrogoths arose from those Goths who remained beyond the Danube and were incorporated into the Hunnic Confederation. With the breakup of that confederation in the mid-5th century, the Ostrogoths became semi-independent players in European politics; they entered the Eastern Roman Empire and became federated. At the invitation of the Eastern Roman Emperor Zeno, Theoderic led the Ostrogoths into Italy in 488, fought a five-year war with the barbarian king Odoacer, murdered him at its conclusion, and reigned for 33 years as a nominal dependent of the Eastern Roman Empire.
The people who would become the Visigoths did not think of themselves as such; the title is one invented by a Roman scholar in Theoderic's service, Cassiodorus, who combined two words, visi and gothi, both of which had been used by or in reference to the people in question. He was apparently under the mistaken impression that the name "visi" meant west, thus Visigothi = the west Goths. At this point, the "Visigoths" had established an independent kingdom spanning much of southern France and most of Spain, so it probably made sense to him.
Tl;Dr: I have no idea why Gotland is named as such, but there were no West Goths and East Goths living in Sweden that we know of.
As /u/Rittermeister said, it is up for debate whether or not the Goths originated from Sweden. If you'll excuse me, I'll quote part of a post I made on a similar matter here:
This isn't my field, but from what I'm reading it seems the difference in intensity of debate is related to the issue of Gothic ethnogensis being tied up in ideas of Swedish national identity (Neville 213-4, 234), whereas the ethnogenesis of the Vandals not only has little evidence but is not tied up in any comparable nationalistic debate. It seems a Swedish historian is probably more likely to argue the Goths are from Sweden or somewhere nearby (Strid 2010) whereas a mainland-European historian might be more likely to argue the Goths were from Central Europe (Kortlandt 2001). This might also be a gross oversimplification of the debate, but you can look at the debate over whether Alexander the Great was Macedonian or could also be considered a "Greek" for an example of how nationalism can complicate historiographical debates.
The evidence for the origins of either is pretty shaky due to the Goths and Vandals both being "prehistoric" in that they did not keep written records before their first contacts with the Roman Empire (Bennett 20). The oral histories of these people as transmitted by important sources such Jordanes' Getica may very well have been fabricated in the first place by the Goths themselves (Bennett 23). Even if we assume that the Goths were a clearly identifiable gens that can be geographically tracked to their ethnogenesis, which is a shaky proposition anyway, the evidence does not exist to give us a precise answer, and the only new evidence we are likely to come by in the form of archaeology may be of no help in determining evidence on ethnicity (Liebeschuetz 210-4).
...Perhaps we will have to make do with the general assertion that the Goths and Vandals were both of "Germanic" origin (Wells 596), whatever that means. Due to limitations of evidence, there are unfortunately questions in ancient history that we can't always answer, and at a glance, this to me seems to be one of them. Jordanes tells us the Goths were from Scandinavia, but the debate over the accuracy of that assertion can probably go in circles forever.
So without establishing that the Goths were from Sweden in the first place, which seems to be basically impossible due to the paucity of evidence, it is unfortunately impossible to answer the question of whether the names of areas of Sweden are related to Gothic groups, that may or may not have lived there, in any way.