There's not exactly complete lore on the multiple alternate universes in League, just some questions that Rioters have answered here and there. I would suggest just going to the wiki to read up on the random tidbits of info that have been compiled.
And for your convenience, I've put the links to the different pages on the wiki here (I just want to feel important):
http://leagueoflegends.wikia.com/wiki/Project
http://leagueoflegends.wikia.com/wiki/Arcade
http://leagueoflegends.wikia.com/wiki/Mecha
http://leagueoflegends.wikia.com/wiki/Pentakill_(Band)
http://leagueoflegends.wikia.com/wiki/Star_Guardian
http://leagueoflegends.wikia.com/wiki/The_Evolution
If you're going to read any of them, read Pentakill. It's surprisingly the one with the most lore.