Switzerland/Petersburg/Pavlovsk; Interior Room/Terrace/Outside; Rogozhin's House; Late 19th Century
We listed all of the geographical locations that are in the novel for a reason: we're not sure we can tell the difference between any of them. Sure, there are some practical differences between what you can do in a resort town versus a small Swiss village, but in terms of description? Well, let's just say that Dostoevsky doesn't really go in for broad landscape painting like his contemporaries Tolstoy and Dickens do.
When you read those guys, you get such a feel for the places they describe that you could easily recreate the book's plot in real life, walking around the streets of London, Moscow, and Petersburg. (And, um, they do. There are plenty of Dickensian walking tours across the pond, for example.) But Dostoevsky is much more interested in the close up than the wide shot, and so there are no particularly evocative views of nature or descriptions of surroundings, even if the characters do pay some lip service to the idea that it's pretty outside.
Which brings to the setting stuff that this book is really made of—the immediate environment each character is in in each scene. If there's one thing Dostoevsky's writing is known for, it's the super claustrophobic, overcrowded rooms where people get together to yell at each other and have insane meltdowns. It's basically like an action movie where all the set pieces are replaced by screaming fights. For this reason, the locations you really come away remembering are the interior rooms where people gather, and where their unwelcome, uninvited, and inappropriate acquaintances show up to get all up in each other's grills.
For example, in the beginning of the novel, there is the great pile up of incoming crazies, as first Nastasya, and then Rogozhin and his crew show up at Ganya's house:
Ganya was motionless with horror.
Nastasia's arrival was a most unexpected and overwhelming event to all parties. In the first place, she had never been before. Up to now she had been so haughty that she had never even asked Ganya to introduce her to his parents. Of late she had not so much as mentioned them. Ganya was partly glad of this; but still he had put it to her debit in the account to be settled after marriage.
He would have borne anything from her rather than this visit. (1.9.1-3)
At this moment there was a terrific bang at the front door, almost enough to break it down. Some most unusual visitor must have arrived. Kolya ran to open. The entrance-hall suddenly became full of noise and people. To judge from the sounds which penetrated to the drawing-room, a number of people had already come in, and the stampede continued. Several voices were talking and shouting at once; others were talking and shouting on the stairs outside; it was evidently a most extraordinary visit that was about to take place. […]
Ganya stood at the door like a block and looked on in silence, putting no obstacle in the way of their entrance, and ten or a dozen men marched in behind Parfen Rogozhin. They were a decidedly mixed-looking collection, and some of them came in in their furs and caps. None of them were quite drunk, but all appeared to be rather excited. (1.10.1-6)
In The Idiot, this kind of party crashing happens constantly, whether or not there is actually a party going on. Think Myshkin and Rogozhin coming uninvited to Nastasya's b-day party, or Nastasya herself making a surprise appearance at the Ivolgins', or Lebedev constantly busting in on Myshkin without any warning.
What's key though, is that all the characters seem to want to preserve whatever small amounts of dignity they have by only losing it inside, away from the neighbors' prying eyes. (They are always commented on, even though Dostoevsky doesn't spend too much time on the idea of rumor or gossip).
Why is this key, you ask? Because many Russian houses back in the day had a terrace or veranda that would be like an additional outdoor room where people would chillax in nice weather. It's kind of in between being indoors and outside. You can have guests hang there, but it's really unlikely that anyone is going to pitch some kind of shrieking fit right out in the open like that. Check out, for example, how Mrs. Epanchin leads Myshkin to the terrace after she finds him meeting Agalya on that green bench, but then doesn't actually yell at him all that much, and instead goes inside. Also, this might explain how much more shocking Ippolit's whole confession shenanigans turn out to be, since he decides to read his manifesto on Myshkin's terrace.
The final immediate place to do stuff is actually outside. What is totally fascinating about this novel is that the way it treats being inside and outside is reversed from the norm. Usually, when we think outside, we think crowds, people, hustle and bustle, right? And indoors, that's where you can go to be alone with your thoughts. But in The Idiot, the last place you can be alone with your thoughts is in your own house, because there is no way to prevent randoms from annoying you with constant, unnecessary visits. So instead, every time Myshkin needs to have a long think about whatever craziness he needs to sort out, he heads for a long walk outside, where he is never in danger of running into anyone.
Sure, not a lot of the novel happens at Rogozhin's house. But, well, the part that does happen, i.e., the murder, is pretty important, no? Not only that, but this house might well be the only physical space that actually gets a lot of description in the novel. It's gloomy, small, strange, depressing, and fills everyone who goes there with a sense of doom and horror. Way to sell it, Dostoevsky. We get it, Rogozhin and his whole house are totally bad news. Why do you think no other place in the novel gets a description like this one? Why don't we get a competing/contrasting image of the Epanchins' house, for example? Why is no other character so closely associated with the place where he/she lives?
Late 19th century
Things were different in the late 19th century and it shows in The Idiot. Russia was ruled by tsars and the serfs where just beginning to revolt, causing some major jitters amongst the aristocracy. Also, as you've undoubtedly noticed, women didn't exactly have equal rights; the bra-burning protests of the sixties (the 1960s that is) were still a good century away. In which ways would The Idiot have been different if it were set today? In which would it have been the same?