You can download Audacity directly from the site here:
Links for Windows (installer), MacOs (installer) and Linux (as source code that you need to compile).
Use this link to get the instruction to install the Lame MP3 encoder :
And download if from here (Windows and Mac) :
Both downloads are executable. Download and run the files. Follow the instructions.
NEVER use the Normal or Default install, always use Custom or Expert and opt out from any and all so called "recommended" components that are never actually recommended by the author of the software that you are installing.