RE-working a site

I have just taken over the duties of maintaining our web site from someone who left our organization. I have built many web sites using Dreamweaver but have never used WordPress before. I would like to continue using WordPress due to the ease of allowing others to edit and update.
I would like to change the theme but like the way the site is layout currently. I was wondering how I can go about editing the site before having it go live? In DW I would just copy all the files to another folder, make all the changes and tests and when ready to go pull them all back.
How do I go about this in WordPress.
Thanks in advance!