There are some really good responses here. Personally, if I were recommending an English translation to a person just starting to read the Bible, I'd say the New International Version (NIV) or, if the person had a slightly more classical background, the New King James. (NKJV)
That being said, it can be both confusing and intimidating for a person to simply pick up the Bible and start reading without knowing a little context. Simply put, most Protestant Bibles have 66 "books" and are divided into two major sections: the Old Testament and the New Testament.
The Old Testament contains many of the stories you may have heard before: the Garden of Eden, Noah and the Flood, David and Goliath, etc. However, many of the stories are not for children. While God has always offered mankind the choice of obedience and a life of justice and peace, the Bible shows that life in the ancient world was often hard and violent. God does not condone evil, but the Bible does not tip-toe around the fact that human beings can do some horrible things. Nor does the Bible paint every main character as being perfect; many of the people that God used were very fallible. Like us, they sinned and then had to repent.
There is another factor that sometimes causes confusion. While there are books of the Old Testament that contain songs/poems, wisdom literature, and prophecy, many of the books relate the history of the Jewish people and the foundation of Ancient Israel. God gave the Jewish people some very specific rules to follow to show that they were different from other people. They were not to eat milk and meat dishes at the same time; they were not to wear clothing made of two different fibers, they were to worship in certain ways, etc. etc. While certain Jewish people still follow some of these rules, most Christians do NOT. Christians believe that many of these practices were symbols or were simply ceremonial. The fact that these "rules" are in the Bible, and yet Christians do not follow many of them, is a stumbling block to some people who do not understand Christian theology.
Some Bibles contain a section of books called the Apocrypha after the Old Testament. These books were not accepted by all Jewish people. Some Christian Bibles have them (although apparently no church considers them on the same level as the other books), and other Bibles don't. Some of them are simply moral stories, at least two are history; there are a few additions to other Biblical books, and there is a book of wisdom.
The New Testament is the story of Jesus of Nazareth, who is accepted by Christians as the Messiah or Christ (a special anointed leader) who was promised to the Jewish people in the Old Testament. Jewish people, of course, do not accept that Jesus of Nazareth was the Messiah. The New Testament has four books that tell about Jesus' ministry on earth (Matthew, Mark, Luke, and John). Another book (Acts) tells about the history of the Early Church. One book (Revelation) is a prophecy about the end of time and the rest are letters written to early Christians and early Churches about doctrine and church-management questions.
When a new Christian starts reading the Bible, he or she is often told to start with the Gospel of John. If you are a voracious reader, however, you can start with the first book, Genesis. Don't let yourself be too bogged down with the genealogies at this point. While they are important, try to see the "big picture" of what the Bible is trying to say.