Your email accounts need to be set up as IMAP or Exchange accounts to do this. Check the providers for each of your email accounts and see if they have IMAP servers and how to setup that way. Then doe so in both Apple Mail and on the iPad.
MobileMe is IMAP.
Google can be IMAP or Exchange
Hotmail can be Exchange (not sure about IMAP)
Also, in Apple Mail your folders will be in several sections. The folders under (On My Mac) are local. The ones listed under your account names are on the servers, and if the server is IMAP, will update on all computers and devices.
the Trash, Sent, Junk, and Drafts folders under MAILBOXES can be set up in preferences, accounts, mailbox behavior.