There are battles between ships and ground battles which play out mostly the same, but with different names for attacks. (thatvideogameblog.com)
Tea parties cost gems, which are also gained by grinding battles. (impulsegamer.com)
We'll update this page as the Mega Stones become more available, but for now we recommend making sure you've acquired every Z Crystal too, and get started on grinding those Battle Royals and Battle Tree for Battle Points! (eurogamer.net)