If it is just old emails that you want to transfer, it may be easiest to buy a USB Jump Drive (sometimes called Flash Drive). You plug the Jump Drive into your old computer's USB port and export the emails to the drive. Then plug the jump drive into your new computer and import the emails into your new system.
You can do the same thing with your Address Book and other files that you may wish to save (like Favorites, for instance).
See more here. I have both a Lexar and a PNY drive and I like both of them very much. The price difference is mostly in the amount of data the unit can store. The more memory, the more cost. I have a 256 MB unit and a 1 GB unit.
If your old computer doesn't have a USB port, you may be able to use floppy disks to accomplish the same thing. Another option would be to put the emails into a folder and use a compression utility like WinZip to compress the folder into a file. Then email that file to your new computer.
Locating the store folder for your email messages depends upon what program that you're using for your emails. If Outlook Express, you can click on the "Tools" menu, then "Options". Click the "Maintenance" tab, then the "Store Folder" button. That will give you the address of your stored messages. Go to the address and copy the folder over to the media you choose to use to transfer the data to your new system.
For the MS Outlook message store folder,
see this page.If you want to continue using your old computer, you could network the two together. The best connection method depends on what type of network card the old computer has. If it has a 10/100 Fast Ethernet card, you can connect them via a switch or even directly.
See more here.There are lots of ways to network computers and that makes this a confusing topic. The Network Wizard in your new computer makes this process pretty simple. Remember, if you're using a Firewall, you will need to give permission for the computers to connect.
I hope this is of some help!
Dwight