Can I install PS3 games on an external hard drive?

You can't install games on external USB drives. Games will only run off of the internal hard drive.

You can replace the internal hard drive. It will take any standard SATA 2.5" (laptop-sized) drive. The drive will be reformatted automatically. See How to upgrade your PS3 hard drive. I upgraded to a 1 TB drive. It took maybe ten minutes to do the swap, but a few hours to backup before the swap and a few more to restore.

You can back up to a USB hard drive, though it takes forever.