Start by checking if you can run Snow Leopard:
Requirements for OS X 10.6 'Snow Leopard'
The OS 10.6 Snow Leopard install DVD is available for $19.99 from the Apple Store:
and in the UK:
but nobody knows for how long it will be available.
When you have installed it, run Software Update to download and install the latest updates for Snow Leopard to bring it up to 10.6.8, or download the combo update from here:
Check via Software Update whether further updates are required. You can now update iTunes in order to sync your iPhone etc.
You should now see the App Store icon, and you now need to set up your account:
To use iCloud you have to upgrade at least to Lion, but some functions are only available in Mountain Lion:
You can also purchase the code to use to download Lion (Lion requires an Intel-based Mac with a Core 2 Duo, i3, i5, i7 or Xeon processor and 2GB of RAM, running the latest version of Snow Leopard), or you can purchase Mountain Lion from the App Store - if you can run that:
It says that you are using OS X 10.5, which means you will need to buy a 10.6 disc from either the Apple Online Store (which will ship it to you) or an Apple Retail Store or Authorized Reseller, and it will cost you around 20 USD.
After purchasing the disc you can then use the App Store to purchase and download 10.8 (another 20 USD or so), which is the newest version. Then you'll be able to update your itunes, as well as enjoy new features that aren't included in your current OS X.
Do not forget to check out the technical specifications to make sure your mac can run OS X 10.8 before you make your purchases and update, and back up your data before you update.