What’s the best way to organize three drives on a Linux desktop?

I’d keep the drives separate rather than combining them with LVM. Put the operating system, applications, and games on the new 1TB SSD. The 500GB SSD is a good place for photos, videos, and active project files. Since the 250GB SSD is very old, only use it for data you can easily replace, such as downloads, temporary files, or a secondary game library.

Move anything important off the external hard drive immediately since it has already reported a failing self-test, and replace it rather than trusting it for storage.

The exact arrangement probably won’t make a dramatic difference for normal desktop use. Games and everyday applications benefit from being on an SSD, but the newer drive’s extra speed may not be noticeable outside of loading and file-transfer-heavy work.

The important part is having backups. Don’t treat separate internal drives as backups if they’re all in the same computer, since one hardware problem or mistake could affect them all.

You could use one of the drives as part of a 3-2-1 backup plan: keep three copies of important data, on at least two different types of storage, with one copy somewhere separate from the computer. For example, keep working files on the 500GB SSD, a local backup on another drive, and an additional backup on external or online storage.

I wouldn’t use the failing external disk for any backup unless it’s only being used temporarily while you migrate its contents.

I’d avoid combining these particular drives into one LVM pool. A pooled setup can be convenient, but it doesn’t make the disks faster, and losing one member can complicate recovery depending on how the volumes are arranged. Separate mount points also make it easier to give each drive a clear purpose and replace an aging disk later.