How to calculate delta v for orbit change around body; not necessarily Earth?

Let me express my joy at having found this great place of Stack Exchange.

My question concerns the calculation of delta v requirements when already in orbit around a body and lowering or raising this orbit as economically as possible (burns at apoapsis or periapsis). For the sake of simplicity, let us ignore atmospheric friction, I would first like to understand the relationship between planetary mass, desired orbit and necessary velocity change.

Expanding further on the question, does this calculation apply equally to orbiting very massive bodies, such as black holes? I would imagine that close to the event horizon, the high speeds of the vessel would lead to friction with space dust that might be accumulated around the accretion disk.