How Do I Get Rid Blue Veins on the Chest?

 See a Board Certified Plastic Surgeon or Board Certified Dermatologist with a vascular laser or a BCVascular surgeon.  They can recommend the best line of therapy.

Visible, blue chest veins can usually be treated with sclerotherapy (injections of sclerosing agents directly into the veins, which leads to their eventual resorption/disappearance). It is difficult to treat these veins with vascular lasers since their diameter/depth is too great for the laser to effectively destroy them.

Thank you for the question! In my practice, I offer sclerotherapy, endovenous laser ablation, Clariven treatment, and microphlebectomy to treat varicose and spider veins. It's not a matter of one treatment being better than the other. Rather, the choice of treatment will depend on a size and location of your veins.  
In recent years, endovenous laser ablation (EVLA) has replaced traditional surgical method of treating varicose veins. This procedure uses laser to heat the veins instead of surgically removing them. The heat produces a steam bubble inside the vein, which destroys the vein walls, causing the affected vein to collapse. The body, in turn, naturally absorbs the dead tissue. 
Sclerotherapy tends to be one of the most popular and widely-used treatments these days. It uses a sclerosing solution that causes the vein to shrink. Usually, sclerotherapy takes only 1 to 2 sessions to achieve fading of the undesirable veins.  
Microphlebectomy is ideally suited to treat superficial bulging veins that are too large for sclerotherapy.  In the course of this treatment, I will make tiny incisions as small as 1mm to remove an undesirable vein.  He will then insert a small phlebectomy hook to extract the vein through the puncture.  Only a small section of the vein needs to be eliminated because the remaining portion of the vein will clot off and gradually fade from view. 
Clarivein is a minimally-invasive procedure that utilizes a thin infusion catheter clinically proven to deliver faster results and guarantee up to 74% less pain than most other peripheral vascular treatments. 
An in-person exam with a board-certified plastic surgeon would be the best way to assess your needs and obtain expert medical advice. Best of luck!