I have an entire chapter in my book titled: SURVIVING EARTHQUAKES, TSUNAMIS, AND OTHER NATURAL DISASTERS.
You should know in advance if you are traveling in an earthquake-prone country. If you are, it is prudent to identify nearby earthquake emergency shelters in advance. These shelters are often pre-identified in tourist areas and hotels. Pay attention to the signs so that you know where to go should the ground start moving.
If you are caught outside a shelter area, cover your head/neck with your arms to protect yourself from falling objects. It goes without saying that you should avoid areas that have heavy objects overhead. Remove anything hanging on the wall near your bed so that you aren’t crushed in your sleep. If you are inside a building, take cover under a very heavy table against an inside wall or in a sturdy (load bearing) doorway. Move away from windows. If you are in a mall or larger department store, stay away from the tall display cases and larger panes of glass. Crowded venues like sporting arenas should be avoided. The panicked crowd will likely cause more injuries than the earthquake itself.
As soon as you recognize that an earthquake may be happening or is immanent, find adequate shelter and gather both your first aid kit and a bright flashlight/headlamp. Those items will likely be the first things that you need should your sheltering structure be damaged by the quake.
If the electricity goes out, don’t use open flames for cooking or for light. There is a good chance that natural gas lines have ruptured in the quake. You really don’t want to blow yourself up.
Expect aftershocks. Some of these will be as bad as the initial earthquake. Be alert for the possibility of flooding or tsunamis after an earthquake. If you are in a low-laying area or coastal region, move to higher ground.
Want some more information? Read my friend Daisy’s article:
How to Survive an Earthquake (and Its Aftermath)