Trim and "keep" and copy move the face ? You can after make some extrude following direction inside the sphere etc...
Depending of what do you want exactly, view inside the sphere or not etc...
Hi Dreamcube017 - yup like Pilou says you want to use the Edit > Trim command for that kind of surface-level cutting. It will slice only the surface of your object up into pieces instead of working at a solid volume level like the booleans are targeted at.
