I'm not sure I understand what you have tried or what you are seeing. Mountain Lion can only be downloaded from the App Store (found in the Applications folder in Mac OS X 10.6.6 or later). Lion is no longer available in the App Store, unless you had already purchased it at some point before Mountain Lion was released, in which case you can re-download it from your purchase history.
First, check and see if your hardware meets the requirements for Mountain Lion:
If it does, you should be able to open the App Store, find Mountain Lion and install it from there. It will begin a 4+ GB download. When that download is complete, the installer should run automatically. If the download is interrupted, that will not happen. You can always re-start the download from the Purchases tab in the App Store.
Just to clarify about the download in the App Store;
You will click the icon for the App - Mountain Lion in this case - and it will take you to that application's page. On the page, near the top left will be the price of the application if it requires purchasing. Click the price and it will say 'buy app', click that and then you should be able to sign in if needed and begin the download/install.