Well, DerMangoJoghurt and Ebonmaw Dragon have already done a lovely job of giving you a good way to get started in on the World of Runeterra, but I'll try helping out too :)
Far, far to the north, are the cold and terrible lands of the Freljord. Here, clans, and clans of clans, battle for survival against each other, and the harshness of the land. Ashe leads one such clan - the Avarosans - and you can read a comic about how she got her start here:
https://universe.leagueoflegends.com/en_US/comic/ashewarmother/issue-1/0/
That should be a good start on the Freljord, but other champs from here include Braum, Lissandra, Sejuani (also in the comic!) and Ornn. There are more.
Slightly to the south, where the weather gets milder and below an enormous mountain range, is Demacia. A temperate land that was once traumatized by ancient magic such that many Demacians fear and hate all magic! This is why Sylas rebelled against Jarvan IV's father, the king, and escaped the capital with a group of oppressed mages. Demacia is otherwise very pastoral, and usually peaceful, and some of the champs from there include Quinn, Shyvanna, Garen and his sister Lux, and Sona.
Here's another comic showing Demacia: https://universe.leagueoflegends.com/en_US/comic/lux/issue-1/0/
Across the land to the east, with various smaller independand states in between, lies the main body of the Empire of Noxus. A huge empire that touches all three known continents, Noxus values strength above all, and takes over territory like it was going out of fashion. They basically don't care who you are, or what you do, as long as you do it with strength, and you do it for the empire. Some champs from here include Swain, Talon, Kled, Sion, and Riven.
It's only a one shot, but it's one of my faves - a comic about Darius in Noxus: https://universe.leagueoflegends.com/en_US/comic/darius/bloodofnoxus/0/
And those three are the main, known, regions of the northern continent - known as Valoran.
To the south, is the continent of Shurima, dominated by the desert region (previously an empire) of Shurima. This mostly desert land was once mighty, with a great river that brought life and water - controlled by the emperors of ancient time and their magic from the enormous Sun Disc magic devices. They and their Ascended warriors (including Nasus and Renekton) conquered pretty much the whole continent before Xerath's betrayal of the Emperor Azir caused the empire to fall. Their people have been struggling to survive ever since. Some champs include Sivir, Rammus, Taliyah, and Kassadin.
On the western coast of the Shuriman continent, around an unnaturally tall mountain range, is Targon. The land is harsh, steep, and deadly - and the people twice as much so. They mostly follow the religion of the sun, led by the Solari (such as Leona) and the very special, rare, chosen few who manage to climb the very highest mountain and reach the special place above, become Aspects - such as Diana, Zoe, Taric, and Pantheon (although, that's complicated now).
On the opposite, eastern, coast of the Shuriman continent are the jungles and arcologies of Ixtal. Hidden from the outside world by geography and magic, mostly they have practiced their abilities and kept away from the (seemingly primitive) outside world for centuries - though Qiana is keen to change that. We've not learned a lot about this area yet, but the surrounding jungles are deadly and full of champs like Nidalee and Zyra.
Across the ocean to the east is the Shadow Isles. Once beautiful, refined, and educated lands that hid from the rest of the world behind their magical white mist, a terrible catastrophe and betrayal caused them to fall into ruin and become cursed. Now, the black mist causes harrowings, and entities such as Thresh, Hecarim, and Kalista haunt and stalk the lands (while entities like Maokai and Yorick try to make things better).
Speaking of the Harrowing (that intermittant arrival of the Black Mist swooping over the ocean and killing all in its path) to the north of the Shadow Isles are the Serpent Isles with the city of Bilgewater. A mishmash of criminals and bounty hunters and monster killers, this tropical, highly vertical, island city builds tall to try to keep away from the ghostly attacks (and the smell of butchered sea monsters). Gangplank once ruled here, but was deposed (violently) by Miss Fortune. Other dangerous sorts that are associated with this place are Fizz, Nautilus, Illaoi, Twisted Fate, and Graves.
If you continue across the ocean even further north, you find our third continent: Ionia.
Ionia is a diverse land of many peoples and many climates - from the dry desert-like southern lands to the frozen north. It's a land of many land masses and islands, of magical forests and of war-scarred coastlines (as the Noxians what happened). The land here has a will of its own sometimes, and the path that should take you home might take you deeper into enchanted marshlands or into trouble with spurned, beautiful, spirits. This is where you'd find Yasuo, Syndra (who has 'borrowed' an island of her own), Shen, Wukong, and Ahri.
Last but not least, on the tiny land bridge between Valoran (the northern continent) and Shurima (the southern continent) are two cities stacked one on top of the other. On the surface, is Piltover the city of progress where the streets are wide and full of technology. Where massive amounts of trade ships pass through their gates in order to cross the land bridge (and thus Piltover makes bank from their taxes). It's a city of new vehicles, of incredible works of public art funded by trade clans, and of gleaming glass rooftops. This is where you'd find champs like Caitlyn, Jayce, Orianna, and Camille.
Below Piltover, in the chasms that formed after the creation of the gates that create all those taxes, is Zaun. Grungy, grimy, and polluted by the chemical poison known as The Grey, the people in Zaun live in the narrow streets and alleyways of the chasm. It's full of industry - of hard working people trying to make a better life - of secretive inventors hiding from prying eyes. It's got enormous glass domes where the rich keep their air-making gardens, and it's got pools and pits of toxic chemicals. This, then, is where you'd find Singed, Warwick, Ekko, Viktor, and Jinx.
(There's also the Void, which isn't really a 'place' you go to - but it's where you'd find the origin point of Kha'Zix, Vel'Koz, and Kog'Maw, and where Kai'sa and Malzahar are known to hang around. And there's Bandle City, the land of the yordles (like Teemo, Tristana, and Lulu) which isn't a place on the map but you can reach there through portals.)
