Here's a post I found from a quick online search. It's been many years since I've done it, but it looks right from what I remember. There are a couple methods, but I'd focus on the simplest (the first option: Backup utility). I will also repost the steps, but there are images in that link that could be helpful, along with some extended discussion.
Your life will be a whole lot easier if you already have an external drive that is large enough for everything you need to transfer. Otherwise you'll probably have to use method 2 and keep track of many smaller devices instead (avoid this if possible). You do not need to use a separate computer or any special software.
Method 1: Using the backup utility
- Go to (Settings) > (System Settings)
- Go to [Backup Utility]
- You will be given three options:
a. Backup: Back-up all data from the PS3 hard disk drive to an external storage device.
b. Restore: Use a back-up that was made to an external storage to restore data to the PS3 hard disk drive.
c. Delete Backup Data: Erase backup data.
- Select the option you wish to use.
Make sure the external storage device has enough space to store all the data from your PS3 system.
Note: Back-ups cannot be made to CD-R or other optical disc media.
Method 2: Manually copying data from the PS3HDD to storage media
Connect your removable storage media (USB storage device, MemoryStick, SD Memory Card, or CompactFlash, etc.) via the appropriate slot/port) to the PS3 system.
From the XMB home menu, you will need to specify which folder you prefer to copy information from (i.e. music, photos, games, videos, etc.)
Once you choose the file you want to copy, highlight the file and press triangle.
This will bring up a menu with three options: Copy, Delete, and Information.
Select the destination. The file will begin to transfer to your removable media. You will be given a status screen that will show you the copy progress.
You will then be taken to a page that states, "Copy Complete".
To verify your transfer was successful, you may highlight the removable storage media, press the Triangle button and select "Display All". This will show you all of the files on the removable storage media. You may also insert your media into another system (PC, PSP® portable entertainment system) and view the files that way.
Whichever method you choose, always double check that everything you want to back up has been backed up. I cannot stress this enough... The last thing you want to find out when your new HDD is all set up and running is that your save files for Duke Nukem Forever weren't copied over.