I have a pain in my wrist that suddenly moved to my hand. No real impact but it's very tender and moving which is concerning... Any ideas?

Maybe tendonitis. What side of the wrist has the pain? Often times patients develop a tendonitis which causes a sharp pain with wrist motion. If it is on the thumb side of the wrist this could be dequervains tendonitis. First line treatment would be using a splint, avoiding provocative activities, and anti-inflammatories (if you can tolerate them).
Hard to say. Hard to guess at this one with small amount of info. If your pain is severe and concerning to you -- and it seems that it is -- it is best that you do get it looked at.
Clarify description. Need to know i f you have swelling, does it hurt to make a fist or grab. Where is it in the hand. Where was it in the wrist. Can this be carpal tunnel syndrome? You need to tell your doctor these answers to have him or her help clarify your problem.