Welcome to the Apple Support Communities
First, open > Software Update to install the most recent version, 10.6.8. Then, App Store will be installed with 10.6.8, so open it and purchase Mountain Lion. Make a backup before upgrading, and see if all your apps are supported > http://www.roaringapps.com
After upgrading, open System Preferences > iCloud, and follow the steps to log in iCloud
Other than accessing email, you cannot use iCloud with Snow Leopard even in its most up-to-date version (10.6.8). The minimum requirement is Lion 10.7.2, but the current OS, Mountain Lion, is recommended.
You can update your Mac to 10.6.8 here:
It's free. This will give you access to the Mac App Store (in Applications) - you use your iTunes login and the credit card registered with the iTunes Store.
You can then purchase Mountain Lion but you must first check whether your Mac is capable of handling it - the requirements are here: