Sign up for a Gmail account
Setup your Gmail account on your current computer and make it an IMAP account
Here are the instructions from Google depending on the email client ( eg outlook,windows live mail,apple mail) you currently use.
E Mail setup instructions for G Mail
Pay particular attention to the advanced settings for the ports
Redirect your current email to Gmail through your isp
This means logging onto for example Bigpond or Adam and redirecting your email address to your Gmail address
You will now have 2 email accounts on your computer your current one and your Gmail
Setup your Gmail account to match the folders for your current account , folders are labels in Gmail
When this is done , go to the folders in your current email account select all the emails in the folder ( ctrlA) and drag them to the corresponding label ( folder ) in Gmail
You can also highlight them all and ( Ctrl C ) and copy them to the label
You will then have moved all your emails to Gmail