How can one tell if God is in favor of a modern value or not?

I love this question. I have wondered this myself and found answers. Here you go, from personal experience and study.

The short of it? Prophets, and prayer.

Prophets

Yes, you are right, the Bible is old, but as Amos 3:7 says, "Surely the Lord God will nothing but he revealeth his secret unto his servants the prophets." God calls prophets so that people can know what He is speaking at the time. That is mainly how we have the Bible, a collection of words of God's prophets, chosen and set apart to lead the people in specific, contemporary issues at the time, and also eternal truths, mainly that of the Atonement of Jesus Christ.

Often, people will being to disobey (apostasy) and not follow God's prophets, so He withdraws them from the earth. When the people are ready again, God calls another prophet to speak His word and clarify truth. This happens all throughout the Old Testament, and a period of apostasy begins again near the end of the New Testament.

So, God just leaves us alone forever? For hundreds of years there were no prophets on the earth, no new revelation. Hence a period of reformation/enlightenment when people sought greater spiritual truths for their day, and to restore teachings of the original Christ-organized Church.

As several denominations (and non-denominational sects too) believe (albeit differently depending on where you go/who you ask), God has prophets in the Church today. But we must be careful not to blindly follow a so-called prophet, ancient or modern. How can you know?

Prayer

God's values, or true eternal principles, will never change. God's truth then is God's truth now.

To know if something is right, or, as we might say, to know God's "opinion" on the subject, we can pray and ask. No promise is more repeated in canonized scripture than "Ask, and ye shall receive." Knock, and it will be opened unto you.

From my own experience I know that God answers prayers and will give you an answer to sincere prayer through the Holy Ghost. One must only have a sincere heart and faith in Christ, with real intent to act on the answer. If you have a question about any moral, doctrinal, or other issue, I know God is there, and that He is interested in you and it. He wants to answer you and He will.

Your question basically boils down to two things: hermeneutics and doctrine.

Hermeneuetics is the study of texts, and Biblical Hermeneutics is the application of textual study to the scripture and the process of determining what it means. This field is relevant to your question because your question itself contains clear trances of not being backed by any kind of hermeneutics at all. For example you mention that a "woman as property" view is condoned by Scripture. This is commonly made assertion made by commentators with an anti-Christian bias, but it is not based on a good reading of the text. In order to come away with an idea like that one must ignore the difference in literary genres between passages. The difference between narrative and prescriptive passages is very important and how that is determined is part of what the field of hermeneutics addresses.

Doctrines are beliefs about any given topic. Good doctrine is the result of careful application of hereneutics and a good overall understanding of Scripture and general revelation. Bad doctrine looks a lot like humanism and is the result of man using his own internal moral compass to delineate what he wants to believe. One must always choose between doctrines.

There are of course conflicting doctrines out there. Some will say that practicing h*m*sexuality is fine for a Christian, others will say it's a result of man's fallen sinful nature and an abomination in God's eyes. In order to determine which bit of doctrine you are going to subscribe to you need to do some research. If, as in the title of your question, you plan on subscribing to whatever you can determine to be God's judgement on the matter then your first order of business is to figure out what he said on the matter in the past and what it means to you today.

This means approaching Scripture using the lens of hermeneutics, examining the Canon to determine which books and teachings you are going to consider and authoritative witness on which to base your doctrinal conclusions, what methods of interpreting the text you will use and then applying that to the whole of Scripture to deduce a cohesive theology (or understanding of God's nature and the nature of his dealings with men). Once you have that you can use it to help guide your discernment on individual decisions. Is X a sin? What do the scriptures I hold true say about the matter, how do I read and apply those scriptures and what conclusions must I draw?

All Christian traditions have some corpus of accepted doctrines. Part of your process in choosing what tradition to align yourself with should be examining their doctrines and the basis for those doctrines to see if their reasoning and sources are sound and comparing them to any other conclusions based on the same sources. When conflicts arise between conflicting doctrines, you should have some clear idea WHY you reject one interpretation over another that has taken into consideration how the various conclusions got drawn.

Lastly, any true search for God's opinion -- his will in any given matter -- is going to include communicating with him on the matter. Although they vary in how they see this implemented practically, all Christians pretty much agree that God has made some provision for guiding people into a proper understanding of his word through his Holy Spirit left as a counselor to men after Jesus left earth. Not asking Him to guide you in your discernment process would be like trying to map out the objects in a dark room without turning on the light.