I once thought that it was amusing to say, "I've never changed my mind before. Does it hurt?" - Evil Roy Slade
And that was true for me for quite a while. I, like my father before me, and his mother before him, was a conservative at birth.
But since then I have changed my mind on a few issues.
Abortion - I was opposed to abortion at first because it is undoubtedly killing people. But it seems to me that abortion is like heroin, both are bad but should not be illegal, maybe. If a woman wants to have an abortion, then odds are that kid's life would start well. On the other hand, that's no excuse to end a life. In any case I much prefer the company of those who are pro-life. I suppose that I have not changed my mind on this issue. Abortion is killing the most innocent.
Capitol Punishment - I once favored capitol punishment because people like serial killers and mass murderers should be offed. But I don't trust the government to competently decide who should be executed.
Border fences - I once favored building a big wall or fence on the boarder with Mexico, becasue we need something to control the boarder. But when boarder walls were used in the past it was used to keep people in. The Berlin Wall and the walls around North Korea keep people in. If the taxes on the rich are doubled, or tippled, to balance the budget (which seems to be what the democrats want), then many people will want to get out. Any hindrance to the ability to leave should be avoided. We should have a more streamlined process for legally allowing immigration and deport more who are tax drain.
Foreign intervention - I once favored a strong, over sized, military; who wants to be weak? But how many of our problems have come from interfering with other countries. Its not really our business anyway. Republicans want little government at home, and big government abroad? Why not arm all citizens with fully automatic weapons like Switzerland? Rand Paul on the subject.
Gay marriage - I don't now, nor have I ever, cared about what your sexual orientation is. (Please don't tell me; I don't want to know.) Gays have existed for more of the history of the world than we have been led to believe (see: ancient Greece and feudal Japan). Kids are undoubtedly best raised by one man and one woman, but a gay couple is probably better than whatever the modern equivalent of an orphanage is. I think that the solution is to not have any laws about marriage. File taxes separately, and have marriages held by whoever wants to perform the non-legally binding ceremony.
I realize that these issues have not been fully thought out, by me, but they are issues on which I have been open to changing my mind.