I've been trying to write a book that takes place somewhere in America, maybe New York, Chicago, or another large city. But the problem is, I live in Hawaii and have never been to a city that large. How should I go about writing about the city? Also, in Hawaii, we use a lot of slang words for talking to people or about things, (for example: we say slippers instead of flip flops) would it mean I would have to start using a different vocabulary for the story?
If you want your story to sound authentic, you must learn and use the slang of the city(ies) in question.
The New York Times had a fascinating dialect quiz a few years ago, and the author just put out a book called Speaking American. I haven't read it, but it sounds like a good place for you to start to grasp how regional dialects differ.
Beyond that, you'll have to travel or spend a lot of time speaking with locals to get the flavor of local language. You may get some from documentaries or news, but real people will give you the best results.
If you want to describe the city properly, same thing: travel or start researching films/documentaries. If you are writing about a real place, you have to experience it to write about it.
It very much depends on why it is New York or Chicago. Do you want the specific flavor of the city? Do you want locals to find your portrayal convincing? If so, Lauren is right. But if the particular flavor of the city is not important, you may be better off treating it as generic American city. There is more that is generic about North American cities than the is particular about them, so this is much easier to do.
Write what you know, or you will be risking to be ridiculed (and deservingly so) by the people who do.
If the location is important to your story, do your research (preferably in person). New York, Chicago, San-Francisco are all big cities, but they are extremely different from one another.
As for the slang--Lauren Ipsum is absolutely right--you better get things right, before new yorkers stone you alive for mispronouncing Greenwich Village or Houston St. (OK, you can get away with pronunciation, using only written words, but you get the drift...)