How do I move photos to iCloud?

Go to Settings > iPhone Storage and follow the suggestions there to manage your storage. You cannot move photos to iCloud and delete them from your iPhone, it will delete them from iCloud and anywhere else they are synced. You can move them to your computer or a storage service such as Dropbox, etc.

Go to Settings > Photos and enable iCloud photos. This will begin the sync process.

To reduce the size of storage on your device, also enable optimize photos. Originals will be stored in iCloud and lower resolution versions will remain on your device.

iCloud syncs, so if you delete a photo on your device it will also delete in iCloud.