For those who might want to go "around the world" in the way I have just done, there are at least two airline-based options.
We used the "One World Alliance." American Airlines is the main United States partner.
A competitive airline group is called the "Star Alliance." Both United Airlines and Continental Airlines are involved in the Star Alliance.
Click on the logos to explore their websites.
And as for exploring "the world," I have concluded that time is the most important ingredient!