To upgrade from OS X 10.5.8, you must first buy Snow Leopard by calling the phone number in the Apple Online Store: 800-MY-APPLE (in the US). Snow Leopard costs $20, or $30 for a five user license. Do not obtain Snow Leopard from any source other than Apple.
Make sure your system meets Snow Leopard's requirements:
To install Snow Leopard for the first time, you must have a Mac with:
- An Intel processor
- An internal or external DVD drive, or DVD or CD Sharing
- At least 1 GB of memory (RAM) (additional RAM is recommended)
- A built-in display or a display connected to an Apple-supplied video card supported by your computer
- At least 5 GB of disk space available, or 7 GB of disk space if you install the developer tools
After you install Snow Leopard you can decide if you want to upgrade OS X to the latest version, Mountain Lion. Mountain Lion is available in the Mac App Store, which is available to you after upgrading to Snow Leopard.
I would not be concerned about those warnings. If a banking website or some other secure system requires an updated browser, it will refuse to work with anything else and you won't have a choice.
You need Snow Leopard for almost everything having to do with iOS devices.
I still use Tiger for a lot of my work. If anyone tries to hack into it, go ahead, make my day.