Greenwichlad - I'd prefer to not publicly announce my location. I'll send you a PM as the building management company owns several buildings in NY and other US cities.

You can also try the following:
http://www.tfc.com/
http://www.rockrose.com/
http://www.stonehengenyc.com/

You might want to see if they can also provide you with a USA credit card.

When I moved to the USA (many moons ago) I was able to obtain a USA credit card before the actual move (by leveraging existing UK bank relationships).

If your employer is affiliated with a Credit Union, it may be worth opening an account there too. CU's are good for your credit history. I even took out a small $2K loan secured with money I deposited in a savings account. Then set up automatic payments for the loan from the savings account. The money went around in circles for a while and it cost me a little interest but it did wonders for establishing a good credit history.

To give you an idea on Rental costs - there are some 2 beds in Maplewood/South Orange/Millburn on the MidTown Direct Train line for $2500 (30-40 min commute to NYC Penn Station. From $14 return).

Typically if you use an agent, upfront costs would be one month advance rent, one and half month security deposit, plus one month agent commission. All required at lease signing. Landlords can require employment verification and credit check.

Do you know where you'll be working? If you're around midtown east, then you should consider Queens (Astoria, Sunnyside, Woodside, Jackson Heights, Forest Hills, Kew Gardens) - you could easily find a 2-bed apartment in your price range and the commute by subway is a bargain compared to London. Alternatively, places on metro-north into grand central could be good. If you're in midtown west, you can look at NJ transit or LIRR trains into Penn station. And if you're downtown, PATH trains from NJ could be good or Brooklyn (but it's expensive - you'd need to go a bit further out with your price range).

it's great that your employer covers your health insurance but check the co-pays and deductibles.

I think credit is less of an issue in NYC where they're more used to foreigners. I didn't have a problem.

The brokers ask for 15% of the annual rent, btw! It's a total con. You need to bargain them down before you start. Don't pay more than a month's rent.

We had an HSBC account in the UK and they didn't want to know here unless you were premium so it's good you managed to do that already. In the end, only Citibank and Chase would let us open accounts as non-US PRs. Like a previous poster, we used the credit union attached to my employer as it was the simplest option.