Hmmmm, looking at the Mint website, it looks like flash should be installed already. It may not be activated in your browser. Which one are you using? Firefox, Opera? It may be that you need to enable it in the browser plug-in section.
Installing programs: Each main distro of Linux has one or more repositories. These are servers that have programs stored for your particular flavor of Linux. One way to install programs in Mint is from the Software Manager. Click the menu button and if you don't see that right away, check in the Administrator or System Tools. From there, you can look under categories like internet or graphics to find what you're looking for. If you know the name like "Firefox", you can type that in the search box. There should be a check box that you would select. Once you do that, there should be a "Apply" button down on the lower right that you would click to start the installation process. Sometimes there are multiple choices for a particular program. In the Firefox example, you might see Firefox dev. tools. You shouldn't need to select those additional programs. The basic install should give you all that you need to get up and running.
On the Adobe link I gave you would go with Linux 64bit, flash player 11.2 for Ubuntu(apt). After you have checked the browser as mentioned above. When you click "download now" it should ask if you want to save or run. Go ahead and run it.