This is one of the first, and for those of us who don't go into medicine, only pieces of anatomy we learn. The blue veins you see on white people's arms look that way because they are carrying deoxygenated blood away from the heart. This is your body's helpful way of color-coding which direction everything is moving for biology textbooks. Blood is red when it has oxygen but blue when it lacks it. And you never bleed blue because blood turns back to red thanks to the oxygen in the air. You've seen even more evidence of it, either in movies or firsthand if you're a homicidal maniac: People who hold their breath or get choked turn a purplish blue before passing out. Because they're not getting any O2! Case. Closed.
When it comes to the color of your blood vessels, your eyes can, and frequently do, deceive the shit out of you. Usually, veins are close to the surface of the skin, and they're the ones that carry oxygenless blood. That means it's true that those blue vessels you're seeing carry blood without oxygen, but the blood itself is not blue. Even the vein itself isn't blue. It primarily looks blue because of the way light reflects off it. The vessels carrying blood toward the heart and the blood they carry are both actually darker (also known as even more) red.
Blood with oxygen on left, without oxygen on right.
They appear blue on white people because of the way light passes through their skin. In typical white-person fashion, whoever came up with this myth looked down at his arm and assumed that the little blue lines running up and down his arms must actually be blue, failing to notice that different colors of skin reflect different light waves, making the veins look anywhere from green to pink when viewed through other colors of skin.
The blue appearance doesn't stop blood from being red, just like the color of the sky doesn't make outer space Carolina blue.