Likely, the greatest issue people have with doctrines such as predestination is that they think it necessarily reduces humans to unthinking robots. Or, better, to inanimate pawns on a chessboard, which God moves around as He sees fit. However, this is a conclusion that is philosophically driven, and not one that is derivative from the Scriptures.
The Bible clearly teaches that people have a genuine volition. That is, they make real decisions, and are really responsible for those choices. People either reject the gospel or they believe it, and when they do either they act in accordance with their will – genuinely.
At the same time, the Bible teaches that all who come to Jesus Christ by faith have been chosen, or predestined, by God to come.
So, there can be a tension in our minds as we try to understand these two concepts. Does God choose me, or do I choose God? And the answer, as unsatisfying as it might sound, is “yes”. A person really believes in Christ, and that is an act of his will. He willingly comes to Jesus.
And yes, God predestined all who come to Jesus by faith.