When you boot up the new PS3, create a user.
Then, go to the PSN sign-up under "Network."
Instead of creating a new PSN user ID, select to use an existing one.
Put in your info (e-mail/password) and you will not have your PSN ID on your new PS3.
You will be able to re-download all your previous PSN purchases under "Downloads" in the PS Store.
Now, for your game saves:
Use an external device (like a thumb drive) to back-up your game saves and move them over to your new PS3.
(Under XMB: Games > Save Game Utility > All the files you want to keep. Select to copy them to the device you have plugged in to the USB) plug the drive in to your new PS3 and do the same steps. Select to copy all the saves from the drive to the Save Game Utility folder.
If you have saves that are "Copy Protected" you will need to have a PS+ subscription, save them to the cloud storage, and then copy them from the cloud storage to your new PS3.
Once you get everything set up on your new PS3, If you want, you can deactivate your account on your old one.
Go here to read how to do this.
I hope this helped. Good luck.