There are a few points which need to be explained regarding Marriage (Nikah) with people of the book:
Firstly
In the Holy Quran, ALLAH has permitted marriage (Nikah) with people of the book, now the question is, what is meant by people of the book?
The Ummah has a consensus that the people of the book refers to those who believe in such a book, which is confirmed as Book of Allah through Quran, such as the Torah (At-Tawraat), the Gospel (Al-Injil), the Psalms (Az-zabur) etc and they declare it as the devine revelation.
Secondly
The vast majority of Christians and Jews today are only so called Christians and Jews, but in reality, they are not convinced in existence of ALLAH nor of any religion, they donot believe in the Torah (At-tawraat) and the Gospel (Al-Injil) as the books of ALLAH, nor do they consider Musa (Moses) and Isa (Jesus) as Prophets and Messengers of ALLAH, therefore these kind of Christians and Jews
are Atheists and have no religion, and by no means can be counted amongst the people of the book.
Thirdly
If a Muslim is married to a woman of the book, then as per Shariah law, the offsprings would be considered as muslim, but the offsprings as a result of marriages with christian women taking place in foreign countries adopt the religion of their mothers. At times the couple decides even prior to getting married that half of the children will be on husbands religion and half on the wife's religion. If such a condition is imposed then a muslim marrying under such condition becomes an apostate as soon as he accepts this condition, because he has accepted his children's infidelity and consented to it, and to be content with someone else's disbelief is disbelief itself. Therefore, after accepting such a condition, this person will be devoid of faith and will become apostate.
So all these things prove that a Muslim's marriage (Nikah) to a Christian and Jewish women as per Shariah is valid, provided that the woman is actually on the Christian or Jew religion and marriage takes place in presence of two witnesses in the Shariah way. But there are lot of chances to be lured into corruptions and temptations in marrying a christian women, therefore it is better to avoid it.