How can I change the sys and system passwords?

DATABASE\INSTANCE
The password for sys, system are visible via the view dba_users.
To change for sys and system (or any other database user):
SQL> alter user <username> identified by "<new_password>";

OEM
The sysman is the actual database user in your OEM repository and will have to be changed from within the OEM itself.
Menu: Configuration -> Edit Local Preferred Credentials

DBASTUDIO (Oracle 8.17)
Menu: Tools -> Connect

PASSWORD FILE (local to system)
The password file is required when attempting to authenticate without operating system authentication being used, for the purpose of starting the database instance (bootstrap). If the database instance is not started, then the database instance cannot be used to perform the authentication.

Go to $ORACLE_HOME\bin and run oradim as follows:

Format: oradim -EDIT -SID <your sid name> -INTPWD <your new password>
Example: oradim -EDIT -SID prod -INTPWD secret

If require remote access use:
orapwd file=PWD<sid_name>.ORA password=<passwd> entries=30