How do I set up both a landing page and front page? Here's the behavior I'm trying to achieve.

If a user types in our domain name www.mysite.com, I want them to go to one page, describing our site.  It can be called www.mysite.com/landing.

But if they hit the logo or the home button anywhere in the site, I want them to go a different page, the one defined in Configuration->Site Information.  It is called www.mysite.com/phront.

Ideas?
It’s as simple as sheena_d already commented on your answer. Define /landing as your home page and link your logo and home button to /phront. Done!