if your transferring data there's 2 methods, one requires a Playstation Plus membership and the use of the Cloud Storage to store your saved games and download them (and your purchases) to the new PS3 console.
the other method i believe should work however i can only assume it'll work based off my work on swapping my ~60 GB Interal Harddrive with a ~300 GB one
Get a External Portable Harddrive
Format the harddrive on your computer to be FAT32, this link may help however the newer your OS is i believe the harder it'll be to format to FAT32. you have no option in this since the PS3 will only recognize FAT32
Use the Backup Utility (Settings > System Settings > Backup Utility > Backup)
Use the restore function on your new PS3 (you should have seen a Restore Option when you was going though to back up)
Sign Into PSN on your new PS3
Register your console (you may need to unregister your old one, not sure what the limit is now), Download your purchases, sync Trophy Data etc.
now i should point out that up until Step 4 i have done, from Step 4 i swapped Internal Harddrives and did the restore so i'm not 100% sure as to what restrictions Sony has added in since i did this a few years ago (buying physical copies of almost every game really saves up on space)
As it is indicated in my second link some stuff wont be copied to the harddrive
The following copy-protected content will not be transferable:Digital Right Management (DRM)
PlayStation® format software game data
PlayStation®2 format software game data
PlayStation®3 format software Game Save Data flagged as unexportable
Trophy data, trophy data is saved onto your Sony Entertainment Network account
so be aware you may loose out on same data however i think the items listed above (with the exception to Trophy Data) can be stored in the cloud with Playstation Plus Membership, but i don't have one so i got no experience with it
Also, be aware that FAT32 has a limitation
FAT32 has a limit of 3.9 Gigabytes per file, which means some high quality HD movies might have to be split into smaller files, and that makes it less than ideal for some HD content.
Source - Step 3
so if something (like a movie) is larger than 3.9 GB you're going to run into a problem however i don't think anything brought from PSN (Games, Movies, Music) would be that big and since external drives need to be FAT32 to be recognized i very much doubt you could get anything that big on it either