If you handle dry ice and your skin gets a little red, treat the frostbite like you would treat a burn. If you touch dry ice and get frostbite such that your skin turns white and you lose sensation, then seek medical attention. Dry ice is cold enough to kill cells and cause serious injury, so treat it with respect and handle it with care.
So What Does Dry Ice Feel Like?Just in case you don't want to touch dry ice, but do want to know how it feels, here's my description of the experience. Touching dry ice is not like touching normal water ice. It is not wet. When you touch it, it feels somewhat like what I would expect really cold styrofoam would feel like... sort of crunchy and dry. You can feel the carbon dioxide sublimating into gas. The air around the dry ice is very cold.
I have also done the 'trick' (which is inadvisable and potentially dangerous, so don't try it) of putting a sliver of dry ice in my mouth to blow carbon dioxide smoke rings with the sublimated gas. The saliva in your mouth has a much higher heat capacity than the skin on your hand, so it isn't as easy to freeze. The dry ice does not stick to your tongue or anything like that. It tastes acidic, sort of like seltzer water.