I get hand and wrist pain when I wash my hands with cold water. Is this normal?

No, may be Raynaud's. Raynaud's phenomenon is a symptom caused by constriction of small blood vessels, usually in the fingers and toes, leading to a lack of nutrients and oxygen going to the fingers and toes. Raynaud's phenomenon can be due to auto-immune disorders, hormonal imbalances, medication effects, etc... A primary care, endocrinology, or rheumatology doctor can look for raynaud's and other possible causes.
Not unless it's. Really cold. You may have raynaud's or beurger's. Arthritis more likely relieved by cold. If you smoke, quitting may resolve the issue for a time at least. If you have one of those diseases above and don't quit, you'll start to lose fingers.

Related Questions

I get a wrist pain when I clean or wash something using cold water or do some work which required hand effort what should I do?

Wrist pain. RICE - rest, ice, compress, elevate. Sounds like a sprained wrist. If not improve, see a doctor to help diagnose.
Conditioning. If you've had trauma to the area, get a careful exam. I suggest a physiatrist (PM&R specialist). If no trauma and you're healthy, you may be overusing the structures there. Do you do repetitive wrist work like computer use? If so, see an ergonomic specialist OT. Also, do some stretching and strengthening to the hand/wrist/arm. A wrist splint during heavy work can also help.