Blue veins on breast & one goin through nipples. They get sore from time to time! What does this mean.?

Growing pains. Breast enlargement either by augmentation or hormonal stimulation can result in enlarged, tender veins in the reticular dermis. They are treatable by a specialist.
Hmm. There is a condition called mondor's disease. This is a painful condition caused by a thrombophlebitis ( blood clot ) of a vein on the breast. It is benign and self limiting. You mention soreness, well it could be... Are you breast feeding, felt any lumps in the breast/arm pit? See you obgyn or a surgeon. Do not neglect!