How can one tell if God is in favor of a modern value or not? The Bible seems to condone slavery and accept wives as their husband's property. But these are generally acknowledged to be immoral in modern civilization.

Yet, there are other modern societal perspectives which differ from their Biblical perspectives that many (if not most) Christians deem to still be ungraceful in God's eyes, like homosexuality.

I'm familiar with arguments that the Bible was written for its own time and pertained only to the era. But, if some values can be progressive and Christian despite what is in the Bible while others cannot, how can one know which ones God is okay with?

Also, is it possible he just doesn't care about stuff like slavery?
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 homosexuality 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.