How to I update from 10.5.8?

I have a iMac bought in late 2008 and currently am on 10.5.8

I have an iPhone 5 with the latest updates but I cannot connect them to my computer because my iMac does not support it.

How do I update?
I tried downloading the new iTunes from Apple but it says my software does not support it.

Thank you
Added (1). Also, when I got to apple and software update on my iMac
it always says no new software is needed.

См. статью: How to I update from 10.5.8?