My experience of NYC in March is that it's freezing. April is much more pleasant. Prices will fluctuate according to seasonality.
Personally, I like being around Times square but it costs. I know many people stay in Brooklyn rather than Manhattan as it's more cost effective, within easy travelling distance and with good restaurant options.
1 week is a long time in Manhattan. It's not that big and has nowhere near the amount of things to see and do as London, Paris, Tokyo and a whole list of other big cities.