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If you want a different look on your homepage, you merely need to make a “Page” and give that a special template (e.g. with no header, etc.). Then select that Page to be your homepage.
Your homepage can be any Page you select. You just need to look into “page templates” or get a theme that has a page template design you like.
The following solution used to work, but I don’t know if it does anymore (someone in the comments said this plugin isn’t being updated anymore):http://wpmu.org/wordpress-different-theme-per-page-post/
If you’re looking for a landing page as your homepage, you can also look into landing page plugins.