How to make the whites of your eyeballs really white without those vessles showing?

Careful here. There normally are conjunctival vessels on the white part of the eye which are needed to bring the normal blood and nutrients to that part of the eye. If your eye is "blood shot" then it can be due to problems such as infection, inflammation, dryness, etc. Treating the underlying problem is key. Avoid over the counter vasoconstrictors as chronic usage has side effects including worse redness.
White eyes. Redness of the eyes can be treated with lubrication drops, get-the-red out drops occasionally, low dose topical steroids, oral tetracyclines in some cases, or other therapies. Some redness is normal though so some redness may persist despite treatment.