There are two recommended ways to achieve this:
Shared Database: If all of the computers are on the same local network and you can set up a central server for holding the data then this is the best option. Geneious integrates seamlessly with the server to store your data centrally, giving you access from anywhere on the network and allowing you to share data with colleagues. See the Geneious User Manual for instructions on how to do this. The manual for your copy of Geneious can be found by going to Help→Online Resources→Download Manual.
External Drive (eg. USB): If you need a simple solution that just works for you then this is the way to go. Storing your data on a USB drive may make Geneious slower but is generally safe as long as you shutdown Geneious and safely eject the drive before disconnecting the drive. To use an external drive for your data, open Tools → Preferences in Geneious and change the Data Storage Location to a new folder on your drive.
Other, not recommended options: For reasons described in "Can I store my database on an external drive or a network drive?", storing your database on network drives or cloud syncing services is not recommended as a way to access it from multiple machines. However, you can export documents in .geneious format to these drives, which can then be dragged into another local database on another machine.