My hands and legs hurt when they are exposed to cold temperatures. Is this normal?

Yes and no. . Small arteries in the fingers and toes normally constrict when exposed to cold and can be uncomfortable but if you experience severe pain or develop ulcers, this may indicate small vessel vasospastic or inflammatory disease (i.e. Raynaud's or buerger's disease (though the latter is usually associated with smoking)).You should see your primary care physician for evaluation, especially if severe.
It's called... ...Raynaud's phenomenon. Normal for some people, but if it appeared recently, may be a sign of an illness. See your dr if this is not something you had all of your life.