We just moved to a new home that is about 3500 square feet and has a basement. I know part of it, is just getting used to the new sounds. However, our heater, when it shuts off, makes a huge 'BOOM' sound! Is that just the way it goes?

Also, after the heater shuts off... the fan must still be running to cool off the heater or something... and the air in the vents makes a whistling sound in the ducts near the ceiling. Can I cover those?

No, don't cover the vents, are they all open 30% or more? Boom noise is probably a contracting duct some where. It needs a suport or better cross break. Also check your air filter.