I want to totally redesign the homepage of my website and use a totally different template from wordpress. How to I go about keeping my homepage exactly the way it is while I make a new one?
I would not do any work on the active site. you can just install a demo site, install the theme and work on that. Better you install Wordpress in your own laptop/computer.
Develop it on offline first. Or you can use subdomain that can only accessed privately. In there you can test the new site.
Easiest thing to do is to have an online sandbox of some sort where you can develop it. Installing and testing on a local machine is kind of a pain if you don't have php and mySQL set up already, so definitely do it online in a password protected directory. Then, when you're ready to go live it's just a matter of deleting your current site and transferring the new one over.