What can I do to make the visible vains underneath my eyes go away?

Laser work. Every metropolitan area has a few physicians who specialize in laser treatments for this sort of thing. However, make sure this is something you really want to do rather than something you feel you need to do or should do -- it's your body and only you know what's right. If there is someone special in your life, make the decision together.
Eyeball or eyelid. If the veins are on the underneath part of the white of the eye, then not much. There are a few eye surgeons that can cosmetically treat this, but its not considered standard of care. If you mean below the eyelid on the skin, then the answer is photofacial with intense pulse light (ipl). Small, superficial spiderveins on the face can be treated with this machine. It is considered cosmetic.