In American cities when on a budget then going to a local neighborhood diner is usually the best option. You can eat well and pretty cheaply at say $10 to $15 for breakfast and lunch. The portions are huge compared to what we are used to here in the UK. I also think you get to try out a real slice of Americana. Getting to sample dishes that you wont see in the UK. I love my meatloaf, mashed potato and greens or a chicken fried steak with country gravy and biscuits.
Whilst in NYC pick up a copy of Time Out NY (like the London original) it is a good way to find out what is going on in the city. What deals and freebies etc can be had. The NY Subway is easy to use and very cheap. You can either get a prepay metrocard or do as I do and just buy a 7 day metrocard.
Oh and make sure that you have an unlocked mobile phone with you and then pick up a prepay sim card. Save on a very nasty surprise of a telephone bill when you get back. They can be surprisingly easy to rack up without even noticing.