It depends on what you mean when you say "redo the site's look and feel".
If you are only talking about the theme and not the posts, pages, comments, and blogroll links: You should be able to create a ZIP file archive of just your new theme directory and install that onto your live WordPress installation via the theme uploader on the dashboard.
The caveat is that the theme needs to be in a new directory that does not exist on the live site, and the theme name does not conflict with any existing theme.
If it's more than just the theme then backup and restore is a good way to go.
In either case just make sure you have a complete file and database backup of the live site before you apply any new changes.
http://codex.wordpress.org/WordPress_Backups
http://codex.wordpress.org/Backing_Up_Your_Database
http://codex.wordpress.org/Restoring_Your_Database_From_Backup
Knowing how to put is all back the way it was is your best safety net.