I'm going to try to be honest (and therefore controversial) on this; I tended to get asked about it a lot when I was a science teacher.
There are a few things it's important to say to start with before I focus on individual areas and why I think what I think in later posts.
Firstly, I think it's very important to say that I think that God knew what he was doing when he got people to write the Bible. So if the Bible teaches that the world was created 6000 years ago over a period of 144 hours, then that's what I believe happened. But I'm not at all convinced that that's what the Bible teaches (and I'll discuss why later). I don't think that the Bible tells us whether the world was created 6000 years ago or 14 billion years ago.
Secondly, I think it's important to say that science is a valid method, and comes up with valid answers to valid questions when done properly. Lots of religions don't teach that, but Christianity does. The traditional way of thinking about it is saying that we can know things via the book of revelation (the Bible) or the book of nature, and both of them come from God. So if the scientific evidence pointed unambiguously to the world starting 14 billion years ago (or 4.5 billion years ago, depending on whether “world” means the universe or the Earth), then that's what I believe happened. But I'm not at all convinced that the evidence for an old earth is as unambiguous as it's often presented (and I'll discuss why later). I don't think that science tells us clearly whether the universe came into existence 14 billion years ago or in the last 20 000 years.
What do I do when what I'm sure science says and what I'm sure the Bible says disagree? I'll deal with that if it ever happens, and it hasn't done so yet.
When I discuss the issue, I usually try to argue people into the middle.
For my part, I've read lots of the arguments on all four sides (old and young universe from the Bible and from science) and I think all of them display cognitive bias and none of them make their case well enough for me to agree with them. I don't think any of them deserve to win...