Downloading a file to install is the Windows way of doing things. In linux we use repositories, the instructions for ubuntu (which Mint relies on) are at: http://www.winehq.org/download/deb
However, adding repositories is one of the most crucial areas that Mint differs from ubuntu, so you'll want to follow these (derived) instructions instead:
1) Open Synaptic Package Manager
2) Go to Settings --> Repositories
3) Click New
4) set the uri to "http://wine.budgetdedicated.com/apt"
5) add the distribution line, depending on which version of mint you're using:
Mint --> Distro line
Gloria --> jaunty
Felicia --> intrepid
Elyssa --> hardy
6) set the section to "main"
7) click reload, then install wine
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