Well, generally speaking, when you see the veins in your eyes, it may means that your eyes are irritated by something. And according to some researches, the main cause of the appearance of red veins in the eye is the irritation, dilation, and swelling of the blood vessels within. That is why you can see veins in your eyes. For example, Dry air can cause irritation of the eyes which often results in prominent red veins in the eyes. Also, fatigue and stress can be the causes too. So you need to go and see the eye doctor and he will have a check for you.
It is normal for have veins in the white part of your eyes, but not your cornea (the clear front part). Have an excessive amount of veins can be caused by fatigue or stay up at night, have a good rest and it could be disappear after a few days. And if not, see a doctor and have a test because it also can be caused by some serious disease.
Your eyes may get infection because of the invisible bacterium. This invisible bacterium may be affected in the bad environment, like the dust environment or others. You may use the eye drops with anti-inflammation to release the symptom to make the eyes get clear. You could also use the warm compress to make the eyes feel comfortable.