Notice that a holiday in a small part of the US of A made a big difference, since the whole country is marked red after visiting only 2 states:
The fastest way to fill my map would be to make some strategical visits to Canada, Russia, Greenland, Brazil or China.
The other way would be to go for the biggest number of countries, and in that case traveling through Europe is the way to go:
(Hat tip: hightext)
[Update 2004.02.23]: Added USA and Europe maps.
[Update 2004.09.27]: Added Cyprus and Thailand.