I advise that you seek out someone who is board certified in performing vein treatments / procedures. Usually for spider veins, this would be a vascular surgeon or dermatologist who runs a vein clinic or vein center. For varicose veins, you are best served by seeing a vascular surgeons, as often there is a deeper problem in the leg that needs to be addressed before the sclerotherapy injections are done. Board certified vascular surgeons receive formal medical and surgical training in treating these deeper vein problems with catheter treatments (such as EVLT laser or Venefit Radiofrequency Closure) and can also offer you the appropriate surgery or operation if that is necessary. These treatment cure the underlying cause, known as venous reflux disease.
The majority of vein clinics, dermatologist, and plastic surgeons cannot offer you the complete effective treatment for varicose veins, but the vascular surgeons can offer you everything that you might need.
With spider veins, there is no underlying circulation problem in the majority of patients, so it would be fine for either a board certified dermatologist or a board certified vascular surgeons to do your treatments. Avoid "spas" that offers vein treatments without a board certified vein MD present.
I hope this information was useful to you.