Honestly, the best way to experience the settings is to play the game, and see how the cards reveal the setting. I find it's a much more magical experience than simply reading a setting book. I like getting snippets here and there. Some art, some quotes of famous texts, some evocative names.
Great stuff.
Quick overview, though. In the beginning there was Dominaria, a huge world. This was the main focus of the first few sets, though there was only a vague sense of what the setting really was.
Since Magic was predicated on the myth of the planeswalker, who goes out, finds magic of the multiverse, and uses it to battle his opponents, they started to branch out to other planes. First was Phyrexia, a rumored realm where mechanical monstrosities lurked, but it was not the focus of a set, merely hinted at. Then came the Homelands setting, which is apparently where Sengir and Serra came from (two iconic names from the world of Magic -- one a vampire, one an angel).
Then came Rath, a plane where an evil warlord was building an army for an invasion. For a while there was an attempt to have an over-arching storyline, with the cards printed following the travels and conflicts of a group of heroes aboard the airship Weatherlight. The Weatherlight was a cool idea, but the execution was too forced, so you'd have cards that were representing specific, but rather boring, moments of a story, rather than being iconic and epic.
The Weatherlight crew went from Dominaria, to Rath, to a shadow plane, to Mercadia, then back to Dominaria in time to defend against an invasion from Phyrexia. After that, the world basically ended, and for a few sets you had a primal world of the aftermath of that war, which was kinda nifty.
Then the designers decided to take advantage of Magic's idiom, and they cast their net wider, leaving Dominaria and that entire storyline behind, in favor of new worlds.
First was Mirrodin, where creatures grew naturally with metal, the seas were made of mercury, and all manner of strange artifacts vied with natural life in a conflict of man and machine.
Then was Kamigawa, the land of 1000 legends. It was based vaguely on Japanese mythology, with some nice twists and great, evocative characters. Unfortunately the set was somewhat weak competitively, so it's been dissed a bit.
Now comes Ravnica: City of Guilds. It is a massive plane entirely composed of mighty stone skyscrapers, aqueducts, sewers, wild parks, and endless, endless buildings. It takes a really cool tack at setting and rules, using some of the core elements of Magic game design, and manifesting them into the setting in a way that manages to seem wholly organic. It's quite cool.
Now if only I had money to spend on it. *sheepish grin*
