Hi! I have Windows XP, and when I click to download, everything works fine.
Open and Save come up and I can successfully download those…
But, nothing ever starts installing.
Any help would be appreciated. 
Please try installing the Microsoft Redistributable Package as per these instructions .
Please let us know if that helps.
Gale
If all you are doing is saving the downloaded file, nothing will start automatically. Assuming you have downloaded the installer.exe, you now need to double-click that file to start the installation process. I have no experience with the .zip file download but suspect from what I have read elsewhere on this forum that it is not a “self-extracting” zip. You will need to follow the appropriate instructions for that zip file.
Some web browsers (for example, Internet Explorer) give you the option on left-clicking an installer or other .exe file to “Run” the file instead of “Save” it. Unless you choose “Run” in such a browser, you will as PGA says have to double-click the downloaded .exe file to start installation, if that is what your issue is.
We do not recommend running the Audacity installer from the browser because some security products block installers or other .exe files running from inside the computer’s temporary folders.
Our zip downloads are indeed not self-extracting. Most modern operating systems will give you an option to unpack or extract the zip file if you right-click over the zip file once you have downloaded it.
Gale