Q:

Black hole gravity is strong enough to trap light. So exactly how hard would the gravity pull on you and a spaceship near the core?

A:

The matter falling toward the singularity inside a black hole would indeed acquire higher and higher energies, and its speed would approach, but not exceed the speed of light.

For more information see the related questions at:
http://chandra.harvard.edu/resources/faq/black_hole/bhole-31.html
http://chandra.harvard.edu/resources/faq/black_hole/bhole-19.html.