I have a bulging varicose vein on my leg, how do I get rid of it besides surgery?

Various treatments. Treatment of varicose veins begins with a consultation with a vein specialist/phlebologist. A history, physical and duplex ultrasound are critical to understanding your particular pattern of vein problems. A treatment plan is devised based on this information, and could include endovenous ablation, phlebotomy &/or sclerotherapy. These are effective minimally-invasive alternatives to surgery.
No more surgery. As dr. Zimmett said, there are a number of good treatment options for bulging varicose veins that don't require surgery any more. The first thing to do is to see a vein specialist/phlebologist. You need to have a standing venous ultrasound exam. The results will then determine what treatment options are available to you.
Choices. Injections are an option, surgery and laser treatments are other options. Your choices will depend on the reasons you have these veins. See a vascular surgeon or plastic surgeon to discuss your options.