I have Mac OS X Version 10.5.8

The first step in Upgrading... is to Snow Leopard = OS X 10.6.x

It is Not available as a download... It is a Paid Upgrade.

Do this first...

Check that your Mac meets the System Requirements for Snow Leopard...

Snow Leopard Tech Specs

http://support.apple.com/kb/SP575

If so... Purchase a Snow Leopard Install Disc...

http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard

Other countries...

http://support.apple.com/kb/HE57

After the Successful Install, run Software Update to get the latest updates for Snow Leopard.

Be sure to make a Backup of your Current System Before Upgrading...

Welcome to the Apple Support Communities

Snow Leopard is a paid upgrade, so you have to buy it on the Apple Online Store > http://store.apple.com/us/product/MC573/mac-os-x-106-snow-leopard

First, make a Time Machine backup of the hard disk. Then, insert the DVD and upgrade. Finally, open  > Software Update to install the most recent version. iTunes 11 needs Mac OS X 10.6.8, so you will be able to use it and sync the iPhone 5 with the Mac