Why is my car creaking while turning and accelerating?

I have a 2007 Honda Civic, which has been otherwise perfect except for one thing: every time I accelerate from a stopped position and turn the wheel right or left, It will make a creaking/clicking noise for a couple seconds until it gets up to speed. It drives and turns perfectly fine after that. Why is my car doing this?

Sounds like the front-drive CV joints are wearing-out.

If you're getting answers of "you need to replace part X," they're wrong. There's no way to tell from your description which part is in need of replacement.

If you take this car to a mechanic, they will either attempt to properly diagnose the problem, or they will tell you that your doohickey joints and whatever rods are worn, and will advise you to replace everything. The correct answer is the first one; don't fix what isn't broke.

Here is what to do: First, go to the auto parts store and buy a $5 mechanic's stethoscope. This is like a doctor's stethoscope, except it has a long metal rod protruding from the end. If you're lucky, your car should make the creaking sound while it is sitting still. If this is the case, have a friend turn the wheel for you while you probe each joint, socket, and whatever other moving part you can get your hands on with the stethoscope. Once you hit on the problematic part, it will be very very obvious. That's the culprit right there.

It is somewhat likely that your car won't want to make the noise while it's not accelerating. This is because accelerating is putting an extra load on the suspension, causing the problematic areas of the part to rub and squeak. If you have a fat friend, this is a good time to buy him a pizza in exchange for his help. Have him artificially load the suspension with his… Mass, and turn the wheel again. If no sound is heard, have him use his heft to push reaward on the vehicle while you test again. Hopefully, some combination of pressure and wheel turning will duplicate the problem. If not, a mehcanic might have to diagnose it for you, which can be costly and is not always 100% trustworthy.

If you can identify the trouble, you can get macho and try to DIY it, or you can go to the mechanic and tell him what the issue is and have him fix it. If it's just the noise that bothers you, a little grease will probably quiet it for a while. However, these squeaks develop when the lubricant breaks down and water contacts metal parts. Rust and corrosion happens, and when the surfaces rub, you get your awful squeaking sounds. For safety purposes, it is best to replace squeaky suspension parts, but budget sometimes gets the final say. Ask your mechanic how serious the issue is. If it's life-or-death, he'll be able to tell you.

Creaking noises normally caused by dry joints or bushes.
clicking normally by CV joints worn, more noticeable when turning and accelerating at same time.

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It could be ball joints or your tie rod ends. Check out these two videos that will show you how to diagnose that creaking noise (and you can listen and compare noises)
How to tell if your ball joints are bad

How to tell if your tie rod is bad