[logicalmonster]

This is somewhat anecdotal, but the immigrants I know who have made the most successful linguistic transition to America did so by watching a ton of American media: watching TV + movies and listening to music. See how English-speaking people speak and practice imitating them. Practice imitating some of the most well-known movie quotes. And try and talk to people if you can.

Offhand, it might sound dumb and a little dated now, but a show that I think shows off a lot of interesting speech patterns to check out is Buffy the Vampire Slayer: there all kinds of English-speaking accents and very interesting colloquial style speech patterns there.