Outsource my email server to Google
Solution 1:
I did the same thing recently for two of my clients. Commander Keen is correct you are looking for Google Applications. There is a free and paid plan. The paid plan has more uptime guarantees and vastly more space.
You route mail from your server to Google by modifying your Zone Record/DNS. Its not that hard to do. And they provide some decent documentation.
I've been quite happy with this arrangement as my server is not bogged down with Qmail, SpamAssasin, Horde, etc.
Solution 2:
You're mixing the terms a bit. Google App Engine is for developing your own services.
You want to sign up for Google Applications: http://www.google.com/apps/intl/en/business/index.html
When doing that, you'll get good instructions on how to relay your email through google. Both in- and outgoing.