Sorry if this seems silly but I am new to WP. I realise that Index template is the most important file. Is it ok to have Index as your "home" page, and can you just assign the Index template to the home page when setting it up in the back-end?
Use home.php just as Sisir said. Copy the content from index.php to that file and you are all set.
home.php tells wordpress that that page should be the homepage in a theme so no backend settings needs to be changed.