Old-earth creationism should not be confused with young-earth creationism (belief the earth is 6,000 years old), theistic (God-directed) evolution, or Intelligent Design (an appeal to a nameless, religion-free designer). The "days" in Genesis 1 refer to long epochs of time and the events described in the early chapters of Genesis reflect real history. The findings of mainstream scientists can generally be trusted, including big bang cosmology, geologic time scales, and radiometric dating methods. There were no dinosaurs on Noah's ark. I am skeptical that large-scale biological evolution provides an adequate explanation for the history of life preserved in the fossil record, but it is still worth studying. Jesus Christ, and not a nameless Designer, is the Creator of the universe.
Can you be a Bible-believing Christian and also believe in "billions of years"?
Absolutely! There is no verse in the Bible that specifically tells us how old the universe is. All the verses that young earth proponents use to support their position can responsibly be interpreted to allow for an ancient universe as well. Visit the "Age of the Earth Resources" page for lots of courses related to this topic.
If you are not a Christian...
Have you resisted accepting Jesus as your Savior because you have heard Christians say things about the Bible that contradict what you learned in science? Young-earth creationism does not represent what all Christians believe. (I think it's questionable whether young-earth creationism is an accurate representation of what the Bible even says.) You can be a real Christian and still believe in an old earth.