First thing, get to know a version control system Subversion, Git, Bazaar, Mercurial are some examples. They are a safety net that can save your bacon because they save EVERY change to EVERY file in your fileset.
Then, typically I have a local development server and also a subdomain (staging.example.com) on the production server. I then do my heavy development on the local development server. Then I use SVN to archive all my site changes. Then, using a shell account on the production server I check out the new version of the software to the staging subdomain. If it works ok there, I can then update the live site using just a single SVN check out.
I've also heard of people placing a symbolic link in the location where the site root should be (/var/www/public_html) that points to the live directory (/var/www/site_ver_01234) , then set up the new version in a parallel directory (/var/www/site_ver_23456). Finally, just recreate the symbolic link pointing to the new version's directory. The switch is instantaneous and transparent. I'm sorry I'm not more clear on this method though, I read about it a while back but never tried it myself though.
I've also looked at Bazaar (another version control system) that has a plugin that automatically ftps any changed files to a given server every time a version is checked in.