I have learned that when I come across those passages there is something that I am missing, some social understanding, or history, or cultural perspective that is key to understanding the passage. Many times the key to understanding a passage is to read it in the original language. But if you are not feeling up to learning Greek and Hebrew I would suggest using one or more different translations of the Bible.
To give an example recently I was having a hard time understanding John 2:23-25. The King James Version reads like this:
23 Now when he was in Jerusalem at the passover, in the feast day, many believed in his name, when they saw the miracles which he did. 24 But Jesus did not commit himself unto them, because he knew all men, 25 And needed not that any should testify of man: for he knew what was in man.The more I thought about this the more it didn't make sense. So I tried the New International Version (NIV).
23 Now while he was in Jerusalem at the Passover Festival, many people saw the signs he was performing and believed in his name. 24 But Jesus would not entrust himself to them, for he knew all people. 25 He did not need any testimony about mankind, for he knew what was in each person.That was a little better but still not very clear. Next I tried the New Revised Standard Version (NRSV).
23 When he was in Jerusalem during the Passover festival, many believed in his name because they saw the signs that he was doing. 24 But Jesus on his part would not entrust himself to them, because he knew all people 25 and needed no one to testify about anyone; for he himself knew what was in everyone.It was slightly different but just similar enough that it didn't clarify it. So I tried the New Living Translation (NLT).
23 Because of the miraculous signs Jesus did in Jerusalem at the Passover celebration, many began to trust in him. 24 But Jesus didn’t trust them, because he knew human nature. 25 No one needed to tell him what mankind is really like.This version really cleared it up and helped me understand this passage. With these different translations I was able to check the original Greek and get a sense of some of the key words to gain a better understanding of what John was talking about.
I once asked a New Testament scholar which translation would be a good one to use if I wanted to use something other than the KJV. Their response basically was, "Well it depends on what you want. I really can't say because they are all so different." I have asked a few other Bible scholars the same question and they all gave me the same answer.
But I'm not a Bible scholar so I can give a straight answer. Use the New Revised Standard Version (NRSV). You can read it online at Biblegateway.com.
Now a more nuanced answer.
If you do not consider yourself to be a scholar and you have only ever used the KJV then start with the New International Version (NIV). You can read it online at Biblehub.com or Biblegateway.com without having to purchase it.
The NIV is a modern translation in the spirit of the KJV. It is done by Evangelical Protestants and has become very popular in Evangelical churches supplanting the KJV as the de facto standard for churches in the US. It is a version that will be a very natural move for most Latter-day Saints in the United States.
The drawback of the NIV is that it doesn't have the poetic feeling of the language of the KJV. There are a few memorable passages ("Though I walk through the valley of the shadow of death") in the KJV and the NIV does a poor job because it skews towards a more literal translation rather than a poetic one ("Even though I walk through the darkest valley").
But in terms of comprehension the NIV greatly enhances our ability to understand what is being said. With updated language it makes it easier to read and to get the concepts. It is a good starter translation for someone who is looking to move beyond the KJV.
But if you are going to buy a physical Bible to use for more intense study then definitely get a NRSV. The NRSV is more widely accepted among English speaking Christians and doesn't appeal to just Evangelicals. It is used by Catholics, Eastern Orthodox, Mainline Protestants, and Evangelicals alike. The translation was done by a broader range of scholars and thus does not have some of the theological peculiarities that are in the NIV.
The NRSV is one that will help expand your understanding of the Bible and your personal scripture study. If you are serious about studying the Bible then you should get the Harper-Collins Study Bible, which is the standard version used in many college classes on the Bible. This is the version that I use.
So if you are reading the Bible like many Latter-day Saints are doing for Sunday School this year then I would suggest getting another translation to help expand your understanding of the Bible.