I always use
skyscanner.net which looks for direct bookings and through the companies like Opodo and is pretty reliable. I have to say though that August isn't the best time to go to NYC - the flights are £150-200 more than flying a month later and it is very hot. Hot by the beach or in the country is OK but hot in a concrete jungle is not nice. The normally slightly ticked off New Yorkers get pretty angry this time of year! If you can slip your schedule a bit, Labor Day is on the 6th September, the flights are around £400 and there are lots of sales on for the holidays.
I've got firends over there and go for thanksgiving week which is really cheap, the sales start on the friday and it's usually cold and crisp.
However, if you're going to go whenever, I really like the chinatown and canal street area for cheap funky clothes stores. I would definitely go up the Empire State Building. I wouldn't bother with the stature of liberty but instead get the Staten Island Ferry which goes right by it for nothing - you just get off at Staten Island and then get back on again for the return trip.
Everything is walkable from about 60th street downwards if you have the time and inclination but you might want to start getting the subway. It's much safer than you think.
A week is more than enough if you're staying in NYC only, even if you get yourself a trip out to Long Isalnd