Linux Mint uses the Debian package management system called APT. There are various front-ends for APT. Software Manager, Package Manager, Update Manager, and others are all just front-ends for APT. So what you install in one is visible in the others, as they all use APT. Software Manager then is the friendliest way to find and install new programs. Programs are easily browsed in categories, there are featured programs, and programs have screenshots, ratings and reviews from other users of Software Manager. Package Manager is lacking all that but on the other hand has various tools and ways for you to repair problems. It's more suitable for advanced users, though there's nothing wrong with using it instead of Software Manager.
As for Wine, just install the package called wine. You may also need winetricks to do "tricks" to get certain Windows programs working. I'm assuming you're familiar with Wine's AppDB
where you can search for a Windows program you are having problems with and see if there are reviews or tips from other Wine users to get the program to work. You'll often see tips using winetricks.