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I'm currently operating OSX Server 10.5.8 on a Mac Pro 3.1.
How do I upgrade to latest server OS?
Apologies in advance if question displays gross ignorance!
Mac Pro, Mac OS X (10.5.8)
Mac Pro 3,1 is also known as Mac Pro (Early 2008).
Per this and this, Mountain Lion Server appears supported, if you have enough disk and physical memory.
You can migrate from that release, but would have to acquire and upgrade to OS X Server 10.6 either from Apple (if that's still available, and you'll almost certainly have to call Apple to get a serial number for it) or maybe a new-old-stock 10.6 server (if you can find a copy of that) if you want to upgrade.
I'd try the call to Apple first, if you really want to try the upgrade and not a migration.
Here is the Lion-related migration and upgrade information, which does reference migrations from as far back as 10.5.8.
One potential way to migrate might be multiple external backups, then use Server Migration Assistant to haul in the old bits.
FWIW and given the vintage of that Mac Pro 3,1 box, I'd seriously consider replacing that box with a Mac Mini Server and maybe some external Thunderbolt storage. Much smaller box, newer, and likely roughly as capable. Given the vintage of your current Mac Pro server, the upcoming Mac Pro probably well exceeds your performance and storage requirements.
There's also an OS X Mavericks release "coming this fall" per Apple announcements.