I realized that I made the error of not giving any examples in my original post. While it can be a little shady, I think we all have a pretty good idea of what truth is in the context of physics. The idea of religious truth is a bit harder to get everyone to agree on. I kind of cringe at the phrase "religious truth"—what I'm trying to point to is something more than religion, but I'm not sure what phrase to use instead. In an LDS context, "gospel truth" would work pretty well, but I think the best phrase to use is "spiritual truth"—it's not quite right, but it will have to do for now. Spiritual truth, then, is something that's a bit more vague. When I was a missionary in Thailand, almost every day someone, in response to us inviting them to learn about our church, would tell us "ศาสนาทุกศาสนาสอนให้เราเป็นคนดี," which translates as "every religion teaches us to be good people." For these people, all religions are true, because they all teach good moral precepts. I see other people who judge how true a faith is by how well it meshes with their own beliefs and feelings. I hope an example will clear up what I meant to refer to when talking about truth in the context of religion.
In 1820, Joseph Smith was fourteen years old and was living in Palmyra, New York. It was a time of religious revival, and he was frustrated by the competing claims of the various denominations that their teachings were correct. Wanting to know which sect he should join, he turned to the Bible and was struck by the direction given by James to those who lack wisdom: ask God (James 1:5). He went out into the woods to pray. As he prayed, two glorious persons--God and Jesus Christ--appeared before him. After Joseph gained his composure, he asked which sect he should join and was directed that he should not join any of them. Three years later, he again prayed for guidance and was visited by an angel, who told him about a record engraven on golden plates that was buried not far from his home, and four years later he was allowed to retrieve the plates. He subsequently translated the engravings through inspiration and the record was published as the Book of Mormon in 1830. That same year Joseph Smith, acting under divine instruction, established the Church of Jesus Christ of Latter-day Saints.
I believe that I know all of this to be true. From my point of view these events either happened, or they didn't. Either God lives and is a glorified personage, or He doesn't and isn't. Either the only way for us to be freed of our sins and become like God is through the atonement of Jesus Christ, or it isn't. Either the Book of Mormon is an ancient record translated through divine assistance, or it's a fraud. I don't see how it's possible to take the middle ground on any of these, besides ignoring them altogether.
There are going to be varying ways in which people choose to respond to these assertions and integrate them into their lives. While the right way to do so may be somewhat subjective, the truth or falsehood of these statements is still objective. Likewise, just because people follow different religious beliefs doesn't mean that they are all equally true (or, more logically, all equally false). (There are some interesting theological issues that come up with this, but suffice it to say that our faith teaches that God is just and no one will be condemned for following the truth that they have.) Going back to physics, just because two research groups measure different values for a fundamental constant doesn't mean that they're both equally wrong, or that the constant isn't in fact a constant.
I guess part of the point that I was trying to make in the original post is that I believe that there is objective truth about God and about what we must do to achieve our highest potential, and that much of this can be reduced to kernels of truth. The truth of these things is just as real as the truth of physics principles, the only difference being that we can't use particle accelerators and torsion balances to test them—we have to test them spiritually.
A friend of mine has done a much better job than I could do at addressing this and related topics on his blog at http://thinkersaccord.blogspot.com/, so go give it a read.