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Of course it's OK to read the KJV along with other translations! I would hope that you never give up reading the KJV. Nothing is quite like it. On the other hand, most people have trouble with the archaic language. Ironically, when the KJV first came out some condemned it for being too simple, too easy to understand. Now it is often revered for its obtuseness and archaisms.
As for the differences between the KJV and modern translations, there are essentially two kinds: translation and text. The textual differences between the KJV and modern translations number about 5000. That is, there are about 5000 places in the Greek text of the New Testament where the KJV translates something different from what modern translations translate. Most of these cannot even be translated, however, so the actual differences in English are significantly smaller. Further, the largest category of differences seem to be due to liturgical emphases throughout church history. That is, as time went on, scribes would copy out portions of manuscripts to be read at certain times during the year. These are known as lectionaries. Some churches today still use lectionaries--e.g., Episcopalians and some Presbyterians. There are passages from much of the New Testament strung together by themes. This is what saints have read of the Bible for centuries. And what the scribes were most used to. Now, when they came to copying continua lecta--or manuscripts that were whole Bible books in their canonical sequence--they were influenced by the lectionary readings. This had a direct impact on what they wrote down in these continua lecta manuscripts. Thus, in Mark 6-8 there are 90 verses in a row in which Jesus is the primary figure and yet he is not identified with any noun! Pronouns carry the narrative. At times, it can get confusing as to who is who (cf. especially the healing of the blind man in Mark 8). The lectionaries, however, would start right in the middle of a text. Think what it would be like to be reading from a passage for the day and it starts out, "Now when he said to them..." Very confusing! It is no wonder that scribes would change such texts in light of the lectionaries to "Now when Jesus said to his disciples..." This kind of thing happens repeatedly and comprises probably the single largest group of differences between the KJV and modern translations. Now some folks argue that modern translations are removing the name of Jesus and are therefore Satanic. This position can only be maintained if the lectionaries are ignored and diabolical motivations are to found in the early manuscripts.
The other major difference is translational. This involves a whole set of things--new evidence as to what the Greek and Hebrew mean, archeological findings that confirm certain spellings and incidents in history, interpretational results from thousands of hours of research in the ancient data, etc. All of this contributes to improved translations. Let me put forth an example. In Romans 3.22, Paul says all are justified by "PISTIS CHRISTOU." This Greek expression has been variously translated as "faith in Christ" (most modern translations), "the faithfulness of Christ" (NET), or "faith of Christ" (KJV). The Greek word PISTIS can mean "faith" or "faithfulness." And CHRISTOU generally means "of Christ" but "of" covers a multitude of semantic possibilities. Does this mean that Christ is faithful and his faithfulness is what saves us? Or does it mean that we have faith in Christ and that is what saves us? Either option is possible grammatically. There are no textual variants that impact the decision. The KJV editors simply punted on the problem, not making any commitment to the meaning of the text. The result is that the KJV is almost non-sensical here. Not quite, but almost. Although most modern translations have "faith in Christ," that is a decision that may be motivated more by the Protestant Reformation and Luther's translation than anything else. In the past three decades more and more New Testament scholars are coming to see "faithfulness of Christ" as the meaning here. Some doctoral dissertations have been written on the subject. Thousands of man-hours have gone into better understanding the biblical text.
At bottom, we are not exactly sure which one of these two options was in Paul's mind. But a translation doesn't have the luxury of listing both options in the text! One goes in the text, and one goes in the footnotes. There are thus many and significant differences among the translations. And these represent thousands upon thousands of hours of labor. At the same time, one can get saved reading virtually any translation. On the big issues, it doesn't matter too much which translation you read. On the smaller issues, it becomes very important. Keep all this in balance as you seek to serve the Lord.