But it doesn't understand if you move from one time zone to
another time zone

I use Google Calendar a lot. You'll find all the Hang Gliding
Competitions on it at http://ozreport.com/calendar.php. And while I don't have a problem with it
for hang gliding competition (because I don't put a time in for them), I do have
a problem with it when Belinda schedules an appointment with a time and I change
the default time zone in the calendar as I travel from one time zone to another.
The time of the appointment changes in my calendar, when in fact the appointment
was for the same time as when Belinda put it in.

The Google calendar doesn't handle time zones and local time correctly. Maybe
they'll fix it at some point.
Here is a recent discussion of this issue.
Here is Google's lame attempt to fix it.
Here is a funny view of the problem.
Here is the Washington Post commenting on the problem.