The moment a disaster hits, your priorities are shelter, information and communication, in that order. For earthquakes: drop, cover under sturdy furniture and hold on; do not run outside during shaking, and afterwards move away from damaged buildings and expect aftershocks. Near coasts, strong or long shaking is itself a tsunami warning, so head uphill without waiting for sirens. For floods, get to upper floors or higher ground and never walk or drive through moving water, which is deceptively lethal. For typhoons and hurricanes, which come with warning, stock water and food early, stay indoors away from windows, and do not go out during the calm eye.
For real-time alerts, most disaster-prone countries run official warning apps and cell broadcast alerts that work on any phone with a local SIM; installing the national emergency app on arrival in places like Japan is standard practice. Hotels know their evacuation points, so ask on check-in in risk zones.
Afterwards, follow instructions from authorities, not rumors. Register with your embassy's traveler program before or during the trip, message family that you are safe early while networks still work, and know that embassies actively assist their citizens in major disasters.