I would recommend that you go to see either your primary care doctor
or your dermatologist
. They should be able to take a look at these areas of skin and quickly diagnose what might be going on.
One possibility would be a condition known as dyshidrotic eczema. In this condition, small blisters form under the skin, about the size of a pin head. These blisters itch, and they sometimes lead to peeling and redness of the skin. They might be the "white bumps" you are seeing. If so, dyshidrotic eczema can be treated by a combination of good skin moisturizing, protecting the hands from water and chemicals, and using a prescription strength steroid cream.
If the white bumps, on the other hand, are larger and on top of rather than under the skin, then they may be warts
or molluscum contagiosum, two conditions that are caused by viral infections of the skin cells. Both conditions tend to resolve on their own, but they can also be treated by freezing or topical medications if they are uncomfortable or if they are a cosmetic problem.
Start by making an office appointment with your doctor at your earliest convenience, and good luck!.