These are the files you need to copy over from your user's Library folder:

for Internet Accounts
.../Accounts (folder)

for Mail
.../Mail (folder)
.../Mail Downloads (folder)
.../Containers/com.apple.mail (folder)
.../Preferences/com.apple.accounts.plist (file)
.../Preferences/com.apple.accountsd.plist (file)
.../Preferences/com.apple.mail-shared.plist (file)
.../Preferences/com.apple.MailMigratorService.plist (file)
.../SyncedPreferences/com.apple.mail-com.apple.mail.vipsenders.plist (file)
.../SyncedPreferences/com.apple.mail.plist (file)

for Keychain
.../Keychains (folder)