In Windows I could open up the Explorer and visit any folder to my heart's content.
In Mac OS, I can't seem to do the same. I open the Finder and I can only see things like Desktop, Applications, Downloads, etc, but I want the entire file system.
If you add your hard drives to the Finder sidebar, you will be able to navigate to the root file system "/" directly in your Finder. Then you will be able to browse the entire file system, because all other file systems are mounted somewhere under the root "/".
Therefore, you are probably looking for this: https://apple.stackexchange.com/questions/72821/how-do-i-make-the-hard-drive-visible-in-the-finder
Simply speaking, you can add your hard drive(s) or "Computer" to the side bar in the preference of Finder. You can go to Finder Preference by opening Finder, and clicking "Finder" on the system menu bar, and take "Preference".
There's a few ways to do this.
Make sure your Toolbar is showing and you have the 'Path' icon on it. From there you can click right back to you Mac and then go forward.
You can also enable your 'Path Bar' (view->Show Path Bar) and you can double-click on higher directories.
You'll also find shortcuts to common directories in your 'Go' menu, along with keyboard shortcuts.
If you want a new folder to show in favourites, just drag it there.