Is it normal to suddenly get purple veins along the side of your fingers or on your palm?

Probably nothing: Normal vein color is purple. If there was no trauma and they are not painful or swollen, you can wait to see if the discoloration fades. If you are in a cold climate, you could get a purple discoloration of the fingers called "Raynaud's". Try elevating your hands above your head. If the veins recede with gravity, then there is nothing to worry about. If they are painful, see a doctor.

Veins: Depends . Some can get in the cold weather and if they are more sensitive to cold. More info and photo will be helpful to explain it more.