I’ve been told a biopsy can be painful. Is this true?

Varies. Biopsies are taken for many reasons and in many areas of the body. Some may involve areas that can be painful while others are quite pain free. A consultation for your answers personally would be indicated.
Possibly. It really depends on the nature of the biopsy, and there are multiple different types. On one end of the spectrum are shave biopsies, for example, where a small blade will shave off a piece of skin or a concerning lesion. On the other hand, there are bone marrow biopsies, where are large needle is placed into the hip bone (typically) to remove bone marrow. This type can be quite painful.