I am also a Linux user. There are several simple ways to do this.
1. You did not mention what type of hard drive the old system uses. Since its running XP the drive could be an older IDE type. It could also be a SATA drive.
In any case remove the old drive before you install Linux Mint. If the new drive is SATA install the new drive to the same connector that the old drive used. If the new drive is some sort of IDE drive set the drive jumpers to the master setting. Do the complete Linux Mint install on the new drive. After the install shut down the system and reconnect the old drive to a second SATA connector. If the original drive was a IDE type set its jumpers to slave and reconnect. Power up the system. Once you've logged into Linux Mint you should be able to access the old drive's data files directly from Linux Mint.
2. An alternate method is to use the Linux Mint DVD in the run from disc mode. Once it reaches the desktop use a usb flash drive to copy any files you need from the original drive. After you have all the files you need from the old drive shut the system down. Install your new drive and install Linux Mint on that drive. Use your usb drive to copy the files from the old drive to the new system.
3. Another simple method is to purchase a usb drive caddy that is appropriate for your old type of drive. Remove the old drive and install it in your new drive caddy. Install your new drive into the system and install Linux Mint. Plug your usb drive caddy in to your new Linux Mint system and copy files.
Welcome to freedom. 
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