This is a good question to pose to your family doctor
Veins that begin to stick out in the legs are usually varicose veins
. Varicose veins occur when pressure in the veins starts to cause the veins to dilated and become less straight and more contorted. Varicose veins can occur in anyone at any age, although they do tend to occur more in the elderly. However, varicose veins have a strong genetic component, so if you have other family members with varicose veins this is also more likely. Finally, varicose veins are much more common in women, especially after childbirth.
If the veins that you are seeing are not twisty and contorted like is typical of varicose veins but are, rather, straight and normal looking (other than that they are sticking out), this might occur in a situation where you have started to exercise more regularly. Runners, for example, often have protruding veins on their legs, mostly because the increased muscle tone pushes the veins out towards the surface of the skin.
Finally, even normal veins become more visible as we age, as the skin can become thinner and more translucent. This may be occurring even in the 30s, especially if you tan or are out in the sun frequently.
In any case, I recommend discussing this issue with your doctor. He or she can perform a full evaluation and make sure there is no risk. Best of luck!