New York !!!
Everyone is offering advice on Chicago. So, I guess I'll help out with New York.
First off, if you're flying to NY, you will be landing at either JFK or LaGuardia. Neither airport is in a great neighborhood. You willl want to take a taxi into the city, for sure. Both airports are OUTSIDE of Manhattan.
See, theres New York, then theres Manhattan. To be near the pulse of the city, Manhattan is where you want to be.
You'll want a hotel somewhere around mid-town. I HIGHLY suggest you research a hotel in advance, for the price.
Some rates will be outright robbery, as normal for midtown, but midtown puts you in the center, and makes getting around alot easier.
Manhattan forays: Including suggested times

Morning:
First off, I highly suggest visiting the World Trade Center monument, which is downtown. I feel anyone visiting the city should feel slightly obligated to visit that.
Late morning: Battery Park. Don't take the ferry TO the statue of liberty, or Ellis Island. Just look from afar. Here, you can see the immensity of the city, the river, the harbor, and more.
Afternoon to late afternoon (or as long as it takes

Go to Greenwich Village. Just a bit north of Battery Park. If you walk, you'll walk a bit, but pass through familiarly named areas of downtown called Soho, Chelsea, Chinatown, and into Greenwich Village.
The Village was Manhattans area for the hippie movement years back, the mecca, and it still is today, and more. There are Jazz clubs, Blues clubs, well, lots of the SMALL clubs one would want to check out.
Theres Washington Square park, where you can see a collection of diverse New Yorkers in their native dress, so to speak. Can be literal OR figurative, depending on the particular day.
Lots of sidewalk musicians.
You can eat virtually anything you can imagine, and then some, anywhere in the city.
Day two: hang around midtown. Theres Central Park. Again, a diverse cross section of New Yorkers can be found here. And the park is beautiful.
The museum of Natural History, the New York public library (worth a glance from inside should you come near it.)
Now, there are tons more to do, but these suggestions will give you a good look at New York.
There are, as well, some DON'T's.
Dont carry your wealth in your wallet. Carry about 20$ american in the wallet, the rest in a pocket. Travellers Checks are the way to go, and if so, STILL split up the wealth. But thats a general statement for all travel, isn't it?
But, back to NY: Dont play any sidewalk games, like 3 card monte, or any other contests that involve cash. Unless, of course, you just have cash to hand a total stranger.
They're professionals, and work in teams. One is the game runner, one is a dumb looking player who happens to be winning as you approach. Hell, he'll even pay for your first round, just to show that you can win. AND hand you the winnings.
DO: Try a slice of NY pizza. Same can be said for Chicago tho
DOnt: take a taxi everywhere, unless its to get somewhere fast. Use the bus, or subway.
Subways are a trip. Its a gotta do. Be on guard though.
Now, in NY, your english 'accent' will go a long way. You can act like Led Zeppelins brother

, and get along just fine. Dont ask why, its just that way.
If you're going to NY alone, stay close to 'home' at night. The city at night is best left to the veterans, to be quite frank. Theres alot to go and do, see, but, you run across 'nightlife' eventually. That may not be a grand thing. You never know.
Enjoy NY. Its definately a trip.