There is no one solution that is optimal. Those sort of questions are meant to be open, allowing you to discuss the alternatives, including the pros and cons. For example:
- Setup dataguard, with the new server as the standby. When you are happy to do the move, switchover gracefully and you are now running on the new environments. This impacts licensing and means both hosts need to be on the same version of the database and platform, but it is very quick.
- Transport Database, like doing transportable tablespace, but for the whole database. Only takes as long as the time it takes to copy the datafiles, but restricts to same endian plaform.
- Transportable tablespace. The piecemeal version of the above. More work, but allows a switch between endians.
- Perform image copy backups from primary. Move to secondary location. Start up database and manually recover with latest archived redo logs. Not dissimilar downtime to Data Guard, but a manual process.
- Export/import. Very slow, but logically very simple and allows you to completely change platform and endian without any messing about.
You get the idea. None are "best". All have pros and cons.