does it mean that every point inside a conductor has a net electric field of zero
Yes. Note that this is only the case for a conductor in an electrostatic situation. When there is current there can be a non-zero electric field inside a conductor.
Because if we teleported a test charge inside a conductor, and the electric field is zero, won't it just stay where it is?
According to Maxwell’s equations it is not possible for a charge to teleport. Charge is conserved, so for charge to appear somewhere there must be a current.
However, what can suddenly appear is a charge dipole. If a dipole were to suddenly appear in a conductor then momentarily the conductor would not be electrostatic. The field would not necessarily be zero.
In response to the non-zero field, by Ohm’s law there would be a current. This current would flow so as to cancel out the dipole, returning the charge and the field back to zero as quickly as the conductor can conduct current.