Is it a sin to change what I believe? Or what if I am just neutral?
Growth means you are going to change your mind about things. "Therefore, laying aside all malice, all deceit, hypocrisy, envy, and all evil speaking, as newborn babes, desire the pure milk of the word, that you may grow thereby, if indeed you have tasted that the Lord is gracious" (I Peter 2:1-3) You are going to see things you misunderstood or just didn't know as you learn God's word and then you make adjustments.
"All Scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness, that the man of God may be complete, thoroughly equipped for every good work" (II Timothy 3:16-17).
We all make mistakes. The shameful thing isn't that we were wrong, but that some refuse to learn and improve.
"The way of a fool is right in his own eyes, but he who heeds counsel is wise" (Proverbs 12:15).
"The wise in heart will receive commands, but a prating fool will fall" (Proverbs 10:8).
"A fool has no delight in understanding, but in expressing his own heart" (Proverbs 18:2).
This doesn't mean you change with every new thing that you learn, but when you learn something, examine it, and decide that what you knew before wasn't correct, then the best thing to do is change.
There are going to be times, though, when you learn something new and it will take some time for the idea to sink in. You need to prod at it a bit, chew on it while you look for possible flaws, see how it fits with what else you know, and then eventually put it into action. That is being careful.
"And He Himself gave some to be apostles, some prophets, some evangelists, and some pastors and teachers, for the equipping of the saints for the work of ministry, for the edifying of the body of Christ, till we all come to the unity of the faith and of the knowledge of the Son of God, to a perfect man, to the measure of the stature of the fullness of Christ; that we should no longer be children, tossed to and fro and carried about with every wind of doctrine, by the trickery of men, in the cunning craftiness of deceitful plotting, but, speaking the truth in love, may grow up in all things into Him who is the head -- Christ -- from whom the whole body, joined and knit together by what every joint supplies, according to the effective working by which every part does its share, causes growth of the body for the edifying of itself in love" (Ephesians 4:11-16).
As a simple example, how else does one even become a Christian, but by changing what he used to believe to begin learning what he ought to believe? "How then shall they call on Him in whom they have not believed? And how shall they believe in Him of whom they have not heard? And how shall they hear without a preacher?" (Romans 10:14). The life of a Christian is one of constant growth; that is why study is such an important part of Christianity (II Timothy 2:15).