Changing Wordpress Theme while current one is live.

I am looking for a tutorial that will show me how to install and troubleshoot a new site design while keeping our current site live. Our site was designed by someone else using a theme from MySiteMyWay and I am looking at some of their newer themes - but possibly may go with a different design co. I've become fairly familiar with customizing our current site by myself, but do not know how to create a new site and then migrate that - replacing the current one - once I've gone through the troubleshooting.

I'm pretty good at picking things up on my own - just hoping to get pointed in the right direction.

Thanks for any help!
Karen