Here's a little block that shows how you can set up exceptions using the two common parent/child errors (-2291 and -2292) and then test specifically for these errors in the exception handler:
declare
e_parent_key_not_found exception;
pragma exception_init(e_parent_key_not_found, -2291);
e_child_record_found exception;
pragma exception_init(e_child_record_found, -2292);
begin
-- enter a child record with an invalid parent id
-- try to delete a parent with associated children and no ON DELETE CASCADE
exception
when e_parent_key_not_found then
dbms_output.put_line( 'The foreign key is invalid' );
when e_child_record_found then
dbms_output.put_line( 'This parent row has associated child rows' );
end;