A permanent / 301 redirect is the best way to update search engines with new filenames for existing pages. It should preserve your rankings for that particular page, and is seen as the safest option for doing so.
In order to achieve this, you would create/edit a .htaccess file in your main www directory. The .htaccess file is simply a regular text file named .htaccess. You can add the redirect line(s) to the bottom of an existing .htaccess file if you have one. The format would be as follows:
To redirect old.html to new.html:
RedirectPermanent /old.html http://www.your_domain.com/new.html
To redirect subdir/old.html to newdir/new.html:
RedirectPermanent /subdir/old.html http://www.your_domain.com/newdir/new.html
The first portion is the web path to the file starting from the main www directory, the second part is an absolute URL to which you would like the aforementioned file redirected.
In this manner you can redirect all of your old files to their corresponding new pages and get your visitors and the search engines where they need to be.
If the URL you are redirecting from contains any spaces, enclose it in double-quotes, like this:
RedirectPermanent "/subdir/old url.html" http://www.your_domain.com/newdir/new.html
If you need to redirect a dynamic URL such as:
Send us an email request at firstname.lastname@example.org
to set up the redirect for you since it's a little more involved.