Most American childeren won't know the words and only begin to learn them in high school. Even many (most?) children from Christian families won't know them. As escoville said, you need to learn them to read Shakespeare in high school, but the modern Bible translations don't use them any more.
KJV (King James Version) uses thee, thou, thine as a matter of course.
NASV (New American Standard Version) uses Thee, Thou, Thine only in prayers and Psalms addressed to God; for "normal" speech it uses modern forms; this reflects usage in fundamentalist and conservative Protestant congregations
NIV (New International Version) does not use thee, thou, thine at all. This is one of the most widely used translations in the US.
Other translations (Good News, Revised Standard, American Standard, New King James, the Message, Jerusalem Bible, New American Bible, etc., etc.) use one of the three formats given above. The third one (no thee, thou, thine) is the most common; the middle one is the least common.
I grew up reading the KJV (my first "adult Bible" at age 8 or 9 was a KJV) and have never had any problem understanding this. However, I had to explain it to most of my friends in high school because they had no idea what was happening. This was in the 1960's, and use of the KJV has declined since then.
Today, I point out to my German students that "Thou hast" (etc.) is exactly analagous to "Du hast" (etc.), and that helps most of them understand the English when they read Shakespeare, so German becomes a vehicle to enlighten English.