Have you ever been in a redder area of the country? If you have, think of how the people act toward each other - on completely not political or social hot button issues. While some will be obnoxious, as is true of any large group, most will be basically good people doing what they think (and have been socialized to think) is the right thing to do - although they will sometimes fall short.
In their case, to understand what they think on political issues, it would be best to actually listen to them speak among themselves. In addition, it is good to listen to what they get as input - from an echo chamber. (The echo chamber defeats many who do want to "double check" something - but they hear the same Heritage and AEI talking points from multiple sources and don't see that they are not independent!) If I did not have the natural links to even passively be at gatherings of conservatives, I think that I would look to discussion boards. However, I would ignore Free Republic and look at maybe Fox News comments. There, I would ignore the more reactionary. I would look for the more thoughtful threads where people here are speaking of their values and the policies they then support.
I think part of the reason for the deep divide is that there have always been two main visions of society in America. The one that leads to liberalism/socialism is one that is communal - that we are all connected and have a governmental responsibility to others. The other is the frontier, rugged individualism that leads to Republican/libertarian views.