I am flying from Halifax, Canada to New York, USA to Tampa, USA. I will be able to go through customs in Halifax. Will I have to recheck my bags at the New York airport?
When you preclear US customs at Halifax Stanfield you will not need to go through customs again when reaching the United States. You arrive at a domestic terminal in the US rather than an international terminal and can proceed directly to your next flight. All you need to do is to make sure the airline checks your bags through to TPA, which they will do if your flights are on a single booking.
The airline should handle your luggage if both flights are part of the same journey on one ticket. If it's two separate tickets then you'll probably have to recheck your bags.