No major hazards. Police are contactable on the emergency number 999.
Though the locals often go barefoot they're experts at it!, it's not recommended beyond sandy beaches due to the sharp coral rocks. Use caution when climbing stairs that connect the lower parts of an island near the sea to the upper part above the cliffs. Some do not have railings on the edge, including platforms. Only the most acrophobic would be uncomfortable with this they're plenty wide enough and not vertically "open", but for children, the blind, and someone who's had too much to drink, the risk is extreme. On the platforms, avoid getting too close to the edge -- especially if you need a rest from climbing.
Motorcycle accidents cause many injuries and fatalities. Driving after dark has additional hazards due to poor visibility.
There is a hospital on Rarotonga, and a smaller one on Aitutaki, and some private medical practices operate in the islands. Medical care is limited on the outer islands. Ambulance emergency is on 998.
Try not to eat snappers, they may give you ciguatera. Mosquitoes are mostly a nuisance, though every few years there is a dengue fever outbreak in the wet season. No malaria, or other serious tropical diseases to worry about, but do take dengue fever seriously during outbreaks.
Though the survey form given on arrival and collected at departure is optional, the airport staff will be very disappointed if you don't complete it. In case you've misplaced it, additional ones are available at the airport at departure.
The Cook Islands inhabitants are not behind the times. They have TV and Internet and they know very well what's going on in the world, so don't patronize them. German tourists on outer islands might be asked about Germany's "dark history", but they know very well that these times have gone a long time ago, and that modern Germany is an industrialized and democratic country.
Respect their religious habits; especially that nearly everything is closed on Sundays with the exception of a few bars and shops.
Contrary to popular belief, the Cook Islands own history doesn't include head hunting but there was a large loss of life during the World War I 1914-1918 fighting with the British against Germany and Central Powers.