How can I update my 2008 iMac to the latest Safari OS?

This iMac has not been used for several years. It has Safari Version 5.0.6 (5533.22.3) and is not compatible with my HP Deskjet 1512, but when I check for new software the messages says my software is up to date. My operating system is Mac OS X Version 10.5.8, but my printer requires 10.6. How do I get this update for my iMac so I can use my printer?

Thanks for any assistance you can provide.

The below information is from About This Mac More Info...

Model Name: iMac

Model Identifier: iMac8,1

Processor Name: Intel Core 2 Duo

Processor Speed: 2.66 GHz

Number Of Processors: 1

Total Number Of Cores: 2

L2 Cache: 6 MB

Memory: 2 GB

Bus Speed: 1.07 GHz

Boot ROM Version: IM81.00C1.B00

SMC Version (system): 1.29f1

Hardware UUID: 31DBDAD0-2434-5391-B80C-CDBFE79438C0