Did the Lord God agree with "an eye for an eye"? In the Old Testament there is a scripture about an eye for an eye law:

[Exodus 21:23~25] And if [any] mischief follow, then thou shalt give life for life, eye for eye, tooth for tooth, hand for hand, foot for foot, burning for burning, wound for wound, stripe for stripe.

However, things changed in the New Testament:

[Mat 5:38~42]Ye have heard that it hath been said, An eye for an eye, and a tooth for a tooth:
But I say unto you, That ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also.And if any man will sue thee at the law, and take away thy coat, let him have [thy] cloke also.And whosoever shall compel thee to go a mile, go with him twain.Give to him that asketh thee, and from him that would borrow of thee turn not thou away.

So it seems the 'an eye for an eye' law was agreed in the Old Testament, but disagreed in the New Testament, a contradiction! 

It confused me. What was the genuine opinion of the Lord God in the end? Or did his mind get changed?
The distinction between the Old Testament and New is that the Old Testament laws were given to the nation of Israel as part of the covenant between God and the people of Israel.  The "eye for an eye" seems harsh to many people, but it was actually a law of restraint.  One eye for one eye--NOT two eyes, one arm and a foot for one eye.  "Burn for burn"--not "whole villages aflame".  The idea was to put the punishment into congruence with the actual offense.  Vengeance would most often greatly exceed this standard.

In the New Testament, Jesus is speaking not to the nation of Israel with regard to the covenant, but to individuals.  He was also speaking while the entire world was on the brink of the inception of the new covenant.

So, Old Testament Covenant applies to the nation of Israel as long as that covenant is in effect.  That means between God and the nation of Israel between the time of Moses and the inception of the new and better covenant--roughly 1600 B.C. to 33 A.D.

Questions like this come up fairly often on this site.  People ask, "Should I sacrifice a lamb or not eat pork or not wear clothing with mixed fabrics?"  The answer is, "Yes, if you are Jewish and are living between 1600 B.C. and 33 A.D."  However, if you are non-Jewish or are alive today, the Old Testament Convent Laws are not binding on you.