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 should now see the App Store icon in iTunes, 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:
First, check your hard drive in Disk Utility using your existing install disk.
Then repair permissions.
Then, for safety's sake, clone your entire hard drive to an external hard drive (you keep a backup don't you?).
Installing or updating Snow Leopard: