Can I delete photos on the old phone after I transfered them to a new one?
Yes.

It looks like you're using iCloud to sync your pictures across your devices.  iCloud is a sync solution, meaning that it keeps all your data in sync across every device you're signed in with your Apple ID.

If you delete them from your old iPhone, you're actually deleting them from iCloud.

All you need to do is sign out of iCloud from your old iPhone 6.  This will remove all your synced data (not just photos, in case you are syncing other content).  If you plan on selling your device, you can then erase all the content and your pictures will still be in iCloud.