If you're account is IMAP you can delve into the Advanced settings of that account under Settings > Mail, Contacts, Calendars; select your account. You should find the ability to have your Trash, Sent, and Draft (at the minimum) folders stored locally or on the server. Choose server. You can also create new folders on the server (either with the iPad or the website) and move emails there for better organization.
Exchange accounts will also allow you to create new synced folders on the server.
An Exchange account should already be saving all mail on the server.
POP accounts can do none of this. If your account is POP, check with your provider and see if they also supply an IMAP server, then switch. Switch all your devices and computer clients to IMAP to maintain maximum compatibility and syncing.