The quickest and (arguably) best way to TouchWizify your smartphone is to download and install a launcher called S Launcher, which is available on Google Play.
Once installed, S Launcher should automatically become your new launcher. If that doesn’t happen, you can make it the default launcher from Settings -> Home (you can also choose another launcher from here, or delete S Launcher if you don’t like it and you think you won’t be using it again).
While S Launcher is highly customizable, it will not completely change the UI of your device (for example, it won’t change your Phone book, the main Settings menu, and any other sub-menus). It will, however, change the aspect of your app drawer.
Besides S Launcher, there are other TouchWiz-like launchers, as well as similar themes made for specific launchers (like Nova and Apex) that you can download at any time. You’ll find them all just by searching for “TouchWiz theme” or “TouchWiz launcher” on Google Play.
P.S.: The basic version of S Launcher is free, but you can also get a premium version with extra features, though you’ll have to pay $3.99 for that.