The most convenient way you can import your emails to Google Apps. Simply you have to download and install the Google email unloader. After that select your desired mailboxes from Microsoft Outlook, Outlook Express or Thunderbird and lets get to upload the emails. If you need more support you go first faq section.
Turing some words from email unloader: “The Google Email Unloader is an open source desktop utility for Microsoft Windows. It uploads email and contacts from desktop email programs (like Microsoft Outlook® ) into your Google Apps mailbox. It preserves information such as sent dates and sender/recipient data, as well as the folder structure used by email programs.”
Follow the instructions Blog and Screen Courtesy To Lifehacker:
Move Email to Gmail with the Google Email Uploader [via lifehacker]
Google Email Uploader [via Google Apps]
Google Email Uploader Launches for Google Apps [via cybernetnews]