I tried to read king jame's bible (well was forced by a christian friend who claimed that I will discover true god .. could not go beyond 3 pages).. I will not try quran..
its very difficult for a person of different heritage/culture to get another culture's religious text, by reading it..
And what about the layers of subtext that quran allegededly contains, how an ordinary person like me will ever get that.. what if I take things literally and do something terrible in the view that if it is written in quran, it must be right?
Well - That's up to you, I am not forcing you - I just said try yourself instead of asking. anyway below are the verses regarding veil which you asked, I am directly quoting what's written in Quran - so, you can draw conclusion as per your intellect
“O you Children of Adam! We have bestowed on you raiment to cover your shame as well as to be an adornment to you. But the raiment of righteousness, that is the best. Such are among the Signs of Allah, that they may receive admonition.” (Quran 7:26)
“And say to the believing women that they should lower their gaze and guard their modesty; that they should not display their beauty and ornaments except what must ordinarily appear therof; that they should draw their veils over their bosoms and not display their beauty except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, or their brothers' sons or their sisters' sons, or their women or the servants whom their right hands possess, or male servants free of physical needs, or small children who have no sense of the shame of sex, and that they should not strike their feet in order to draw attention to their hidden ornaments. And O you Believers, turn you all together towards Allah, that you may attain Bliss.” (Quran 24:31).
Below one you must not assume that it's only for prophet - It's for guidance for believers
“O Prophet, tell your wives and your daughters and the women of the believers to draw their cloaks close round them (when they go abroad). That will be better, so that they may be recognised and not annoyed. Allah is ever Forgiving, Merciful.” (Quran 33:59)