HI,
If the Serial Number shows up then this is not your issue.
You can update to Mavericks from Lion and from Mountain Lion.
In Mountain Lion this means you could be anywhere between OS X 10.8(.0) and OS X 10.8.5
Messages was changed at the OS X 10.8.2 update to require that the Serial Number was present to use in the Auth Token that shows in the Keychain.
From earlier you posted:
I have switched to my gmail, @me, and icloud, and none can send or receive.
Is this your actual Apple ID and the @me.com iCloud link email you got followed by the @iCloud variant ?
Messages does treat these as Separate Apple IDs.
If my suggestion is correct which is being used on the iPhone at the moment ?
8:53 pm Monday; November 11, 2013
iMac 2.5Ghz 5i 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad