To update to Mountain Lion you need the App store which requires Snow Leopard first.
Once Snow Leopard is installed you can then download the Mountain Lion installer from the App store.
Also check your Mac is compatible at http://www.apple.com/osx/specs/
If you have 10.5.8, you have the latest 10.5 software.
To make progress, you now need to BUY 10.6.any -- Like 10.5, it comes on DVD, at the Apple Online store. It is currently "put away", but if you CALL the Online Store support number, they can arrange to sell you a copy.
Once updated to 10.6.6 or later, you gain access to the Mac App Store, and further upgrades are generally by download.
Snow Leopard/10.6.x Requirements...
* Mac computer with an Intel processor
* 1GB of memory (I say 4GB at least, more if you can afford it)
* 5GB of available disk space (I say 30GB at least)
* DVD drive for installation
* Some features require a compatible Internet service provider; fees may apply.
* Some features require Apple’s MobileMe service; fees and terms apply.
Which apps work with Mac OS X 10.6?...
It's been pulled from the online store & Apple Stores, so you have to call Apple to buy it, last I heard.
Call Apple Sales...in the US: 1-800-MY-APPLE. Or Support... 1-800-275-2273