Well I am not sure if NZ has tornado sirens, but that would be the biggest giveaway that a tornado may be coming. Unfortunately tornadoes are one of the most unpredicatable things in nature, and can appear almost out of nowhere and without warning. For instance on May 6, 1999 one of the largest tornadoes ever recorded (save the one that hit Joplin, MO last year) hit in the Oklahoma city/Midwest City, Ok area without any warning. In fact the entire day was filled with clear and sunny skies until the tornado hit.
Tornadoes are associated with large (supercell) thunderstorms that often grow to over 40,000 feet. A column of warm humid air will begin to rise very quickly. How the column of air begins to rotate is not completely understood by scientists, but one way the rotation appears to happen is when winds at two different altitudes blow at two different speeds creating wind shear. For example, a wind at 1000 feet above the surface might blow at 5mph and a wind at 5000 feet might blow at 25mph. This causes a horizontal rotating column of air. If this column gets caught in a supercell updraft, the updraft tightens the spin and it speeds up. The rain or hail in the cloud causes the funnel to rotate towards the ground, thus creating a tornado. (http://eo.ucar.edu/kids/dangerwx/tornado3.htm)
So a good thing to watch out for is rotation in the clouds. If you see this do not hesitate to take shelter immediately. If you live in a house that is on a foundation get to the very middle of the house and make sure you are not around any windows. If you live in a mobile home or trailer home, do not stay there get to the lowest point available and hunker down. Get some blankets and pillows and surrond yourself in them. Also a good thing to have is an emergency kit available that has plenty of water, a flashlight, a radio that can be powered by turning a small generator, and a first aid kit. Even funnel clouds have the potential of ripping roofs right off a house, so do not underestimate these storms what so ever.
I have lived in Oklahoma nearly all my life, and have seen a few tornadoes myself. They are very amazing to see, but very very deadly and powerful. I have also helped in the relief efforts in Joplin, MO and have seen the full effect of these storms. Even though New Zealand is not really a hot spot for tornadoes they are highly un-predictible and can show up virtually anywhere if the conditions are right. So just be careful and always take pre-caution if you think a tornado is imminent.