I very much appreciate your asking the question in terms of both law and values, rather than merely seeking a technical permission. It is often difficult to be Orthodox and an artist, and your commitment to be both with integrity does you credit, and moreover I think it likely that the art produced by that dual commitment is a genuine contribution to Judaism and human culture.
Legally, your issue is actually twofold, and I presume you intended both. The first question is whether it is permitted to enter churches when services are not taking place, and the second is whether it is permitted to admire art and architecture that was created for Christian purposes and still serves those purposes.
My own position is that trinitarianism is not halakhically per se avodah zarah (for the basis of my position please see
http://www.torahleadership.org/archive.php?x=0&y=0&q=Christianity), but that bowing to crucifixes and participating in a ritual that treats G-d as corporeal is avodah zarah. Not all churches are therefore alike.
Let us assume that we are dealing with churches whose ritual involves such practices.
This is a version of a very commonly asked halakhic question, “the Sistine Chapel sh’eilah”. My tradition is that my teacher R. Aharon Lichtenstein refused to enter during a tour of the Vatican, whereas my teacher Dr. Haym Soloveitchik permitted entrance if one made clear that one was visiting as a tourist, e.g. by prominently wearing multiple cameras. In other words, Dr. Soloveitchik treated the issue as one of mar’it ayin, of appearances
My default position is that of R. Lichtenstein, and if you were to ask me for a formal decision, that is what I would tell you. However, I cannot say that one is not entitled to rely on Dr. Soloveitchik, although I would be unhappy if one relied on his position for trivial purposes. Regardless, I think that one can rely on Dr. Soloveitchik’s logic to enter parts of a church other than the sanctuary, and I have permitted this publicly for issues such as blood drives, when there are prominent signs posted indicating a secular purpose for entering.
Turning to the values aspect of your question - if there were nothing important to gain by seeing art, this would be an easy question. But at least some authoritative voices in Jewish tradition recognize that there is value in beauty and truth in art (whether or not one sees beauty as a necessary aspect of great art), and so the issue is one of balancing values. The Summer Beit Midrash of the Center for Modern Torah Leadership, which I have the honor to lead, spent its 2011 session addressing questions such as this. You can find the daily shiur of that session at
http://www.torahleadership.org/archive.php?x=0&y=0&q=%2C2011, and more popular materials at
http://www.torahleadership.org/archive.php?x=0&y=0&q=art.
Perhaps the deeper issue is whether it is proper to deeply study explicitly and intentionally Christian art. Here one can introduce the prohibition of “al tifnu el ho’elilim” – not paying conscious attention to avodah zarah works (see Shabbat 149a) – and certainly much Christian art focuses on and draws its power from the most halakhically problematic of Christian rituals and narratives. I am very influenced by a pianist friend who happily played classical masses and the like – until she heard them played in a cathedral and genuinely understood them, at which point she could no longer play them.
Overall, then - the technical prohibitions can often be overcome or evaded, but that they should be overcome or evaded only for a purpose of great value, and only by someone who honestly acknowledges the genuine power of great Christian art and that its messages are often not compatible with Judaism. There is at least some leeway, but you should utilize that leeway only with great self-knowledge as to what might be gained or lost.
I have addressed this question without putting it in the broader context of Jewish-Christian relations today. But I recognize that refusal to enter church sanctuaries, especially at a time when the Pope and other Christian leaders are respectfully praying in synagogues, can be morally troubling, and seem – not unfairly – as reflecting a lack of respect for the deepest convictions of human beings who otherwise command our admiration and on whom we make moral claims. This is a challenge on a different axis, and beyond the scope of your question, and I will not address it here, but I did not feel comfortable replying without at least referring to that challenge.