When a tornado hits and you're in your house, the safest place to be is generally in a prepared shelter or the basement, where the surrounding earth can protect you. If your house doesn't have a basement, get to an interior room as close to the center of the house as possible. Stay on the lowest floor, away from any windows --- and, contrary to popular myth, don't open the windows to try to equalize the pressure. Shield yourself under a table or some sturdy furniture, and make sure that the floor above you doesn't have any heavy items --- like refrigerators or other large appliances --- that might fall through the floor and crush you.
Larger buildings aren't always safer when a tornado hits --- if the roof is only held up by the exterior walls, damage to those walls by a tornado could collapse the entire structure. If you're in a large building constructed in this fashion, try to find a doorframe or other solid structure that can shield you from debris and falling objects. In other regards, follow the same rules as inside your house --- stay away from windows, get to the lowest level and position yourself close to the ground to present a small target to flying objects.
In an indoor shelter designed for large numbers of people, do your best to make sure that a head count is taken as people enter the shelter --- this will help you to make sure that everyone is safe and accounted for after the danger has passed. Upon exiting the shelter, be observant for possible damaged power lines or gas leaks in the building --- avoid using anything that generates an open flame. According to the Centers for Disease Control and Prevention, as many as half of all tornado-related injuries may occur after the tornado itself.
If you live in an area that's particularly prone to tornadoes, consider creating a tornado preparedness kit to help weather the storm and deal with problems that arise afterward. The National Oceanic and Atmospheric Administration recommends that your kit contain one change of clothing per person, one blanket per person and a 3-day supply of water per person (one gallon each per day). The kit should also include first aid supplies, a battery-powered radio and flashlight, an extra set of keys and some money (or a credit card) to handle any immediate needs.