How does one continue their Christian education once they’ve finished reading the Bible?
The best way is through a formal education at a bible believing college or seminary. There are many good choices on the internet as well, if you can't attend a college in person. You must remember, as you are studying, you need to be living a Christian life, and have an open mind that is able to discern properly the word of God. Your testimony must be your own, not your church's or teacher's.
At the very least, one should become aware of basic biblical hermeneutics, and exegetical principles. I have seen it demonstrated on this board, where members lack of knowledge in these important biblical tools, has led to much error and misunderstanding. We must be able to properly interpret scriptures, in the context in which they were written and meant to be understood by the audience to which they were written.
Bible study with other bible believing Christians is another good way to further your education. Your local congregation should be spending time during the week in teaching. Remember, just because someone in authority at your church says it, does not make it so. READ the scriptures, and find out for yourself.
Obviously, it is not necessary to have a formal education to be saved. But in the words of a popular Casting Crowns song, "to know you [God] is to want to know you more." That is the way I feel. We can know God only in part. He is infinite, therefore, we can never know everything about Him. But God is only too happy to reveal Himself to those who seek Him diligently. I am planning on spending eternity learning more about Him.