I work for a vascular surgeon who specializes in veins. It is somewhat unusual to have extensive spider veins at your age (although we’ve treated patients younger than you with very severe vein problems). Heredity is almost certainly the source (as your statement suggests). Spider veins are often caused by a disease called Venous Reflux Disease. To put it simply the 1-way valves in your legs that are designed to let blood return to the heart against gravity fail and you get increased venous pressure in the veins near the surface of your skin. For some people, this results in the rope-like varicose veins, in others, that pressure balloons out the smaller capillaries and reticular veins and the result is known as spider veins.
Your spider veins will likely get worse over time (especially with a family history and the fact that you’re developing them at such a young age). You may want to find a physician near you that’s a member of the American College of Phelebology (enter your zip to locate one near you). I’m not sure that cosmetic sclerotherapy is appropriate for someone your age, but there is a good chance your larger veins are already beginning to reflux and so they may recommend some conservative treatments to slow the progression of your reflux. Regular walking is good for your veins (prolonged standing without walking is bad), elevating your legs at night as well as getting a prescription for medical compression stockings are all simple, conservative treatments that should be appropriate for someone your age and you should discuss these options with your physician.
Sclerotherapy is the standard treatment for spider veins, be sure to find a physician who knows what they’re doing—there are lots of physicians (Physician’s Assistants, and nurses too) who are doing this very simple injection-based procedure. Unfortunately, because so many of them aren’t vein specialists there are a lot of patients who are getting bad results. Sclerotherapy is a bit like an art form, and if done incorrectly (or sometimes even when it’s done correctly) it can lead to staining under the skin, or scaring. If you do have venous reflux, it’s important to treat the reflux before getting sclerotherapy. Many unsophisticated practices (such as health spas, or some dermatology and plastic surgery practices) only treat the spider veins without treating the underlying condition first. This results in the spider veins coming back again and again. If the source of the reflux is shut down, then new spider veins should be much slower to develop.
There’s a lot of info there, and I could keep going on for forever, but in a nutshell, find a physician in the American College of Phelebology (preferably a Diplomat) and get a consult. Ask if they offer a free consult—many do. Get them to prescribe some medical compression stockings (which come in a wide variety of styles now). If you have any questions I’d be happy to answer them to the best of my ability (I’m not an expert on cellulite—sorry).