First things first, we strongly suggest that you check out the handy map Tolkien drew while writing The Hobbit to get a sense of where all of these places are. This map represents Thorin's grandfather's map of the Lonely Mountain. Pretty handy!
Otherwise, we start out in the "kindly West" (18.19), in the land of the hobbits. The hobbits call the place where Bilbo lives The Hill, while the Took family lives across The Water. In other words, the hobbits are so sheltered that they can only imagine one hill and one body of water. They don't even use proper names to distinguish between different hills and waters. (At least, not yet – the Shire gets a whole host of real names, but not until The Lord of the Rings.)
As Bilbo and the dwarves set out on their adventure, they travel east. They leave behind the pleasant greenness of hobbit country and travel into lands "where people spoke strangely, and sung songs Bilbo had never heard before" (2.23). And as they go farther and farther away from Bilbo's home, they find "dreary hills, rising higher and higher, dark with trees" (2.23). The farther that Bilbo goes into the unknown, the more the landscape reflects his discomfort: suddenly, Bilbo is seeing "old castles with an evil look" (2.23) and, of course, the trolls, Bert, Tom, and Bill.
Still, even though the "Lone-lands" aren't familiar to Bilbo, he's in no true danger yet. The real border between comfortable lands and dangerous lands is helpfully marked by Rivendell, the location of the "Last Homely House" protected by the great lord Elrond. Rivendell is a sheltered valley filled with elves singing and laughing. Yet, while the valley of Rivendell is pleasant and delightful, it's still overshadowed by the Misty Mountains beyond. And Bilbo has to leave it eventually.
Bilbo's naiveté about the landscape of his own world really comes to the fore when he first sets eyes on the Misty Mountains, before he and the company stop in Rivendell. Bilbo asks, "Is that The Mountain?" (3.2). He can't imagine that there's more than one mountain – after all, he comes from a place with The Hill and The Water. But Balin replies dismissively, "Of course not!" (3.3). The Lonely Mountain is still miles and miles away. So we have gone from The Hill in Hobbiton to a whole chain of enormous, threatening mountains.
Once Bilbo comes out the other side of the Misty Mountains (and the goblin tunnels), he takes shelter with Beorn the wild man in a large wooden house near a rock Beorn calls the Carrock. Beorn's place is comfortable and safe, but it's still strange and not home-like: Beorn relies on ponies to serve him, and he eats only honey and bread. The strangeness of Beorn's house proves that we are still definitely in the Wild.
Next up, Mirkwood. Mirkwood is a dark forest filled with suspicious Wood-elves, eerie enchantments, and giant spiders. Still, when Bilbo asks if they can go around it, we find out that the Grey Mountains to the north are filled with "goblins, hobgoblins, and orcs of the worst description," while to the South, "you would get into the land of the Necromancer" (7.145). Hm… So while Mirkwood is not safe, there are other places that are even less so – once you are "over the Edge of the Wild" (7.145), there's no safety left.
The Lonely Mountain is the end of the dwarves' quest. It towers "grim and tall" (11.1) and "danger [broods] in every rock" (11.19). The primary characteristic of this setting (besides gloominess) is that it's incredibly hard to get inside. The dwarves have to wait until precisely the right day before they can unlock the side door and get inside. And once Smaug has been killed, the dwarves barricade all of the doors except the Front Gate so they can defend themselves against Bard and the Elvenking. This space is like the inverse of Hobbiton, where we began our journey: the tunnels of the Lonely Mountain are grim, dark, and reeking of dragon, while Bilbo's home is "a hobbit hole, and that means comfort" (1.1). This opposition between Bilbo's hobbit-hole at the beginning and Thorin's dangerous (but wealthy) dwarf tunnels at the end demonstrate just how far Bilbo has gone over the course of his quest.
When Bilbo returns to The Hill, having seen all the things that he has seen, suddenly his own home under The Hill seems changed. (The estate sale of all of his belongings that Bilbo finds upon his return probably doesn't help!) But his home now holds a sword over the mantelpiece, and he takes "to writing poetry and visiting the elves" (19.29). The changes to Bilbo's home – the sword over the mantelpiece and the chainmail coat in the hallway – only serve to prove his internal development as a character.