Oakleaf has given some excellent advice... so "ditto" on what she said.

Remember that a bike steers more by balance, e.g., leaning, rather than turning the handlebars like you would the steering wheel of a car. You actually start a turn by leaning in that direction, and the bike will naturally steer that way.

Get your bike, helmet, gloves, and maybe an hour of spare time and go find a big empty parking lot. Set up a number of paper cups or maybe coke cans in a line about 20 or so yards apart so that it looks like a slalom course.

Now let's get rolling. First, let's ride a loop around the edge of the lot, getting used to holding a straight line. "Twitch" (turn very slightly) your handlebars to the left, and note that when you do this, the bike leans tot opposite direction, e.g., to the right. It works the other way around, too... if you twitch your bars to the right, the bike leans left. Remember this, 'cause it's a good way to start a turn when you really need to turn in a hurry to either dodge something or make a particularly tight corner. Try this a time or two -- in both directions -- as you circle the lot.

Now let's turn... The traction that you get from your tires depends on how much weight they're bearing. You want your weight evenly distributed on the bike's wheels, so scootch your butt back a little bit on the saddle, bend your elbows, and lower your torso a little bit toward the top tube. If you sit up while cornering you could have too much weight on the rear wheel, causing it to slip out from under you on a wet or slick road; likewise if you lean to far forward and put too much weight on the front wheel, it could do the same.

Start your turn by easing the bars in the opposite direction to lean your bike in the direction you want to go. Once you start to turn put the outside pedal down (to the bottom of the pedal stroke), raise your hips slightly off the saddle, and shift your body weight to the pedal. It should feel like you're standing on the pedal with your outside leg. This weight shifting lowers your center of gravity and makes your bike more stable in the turn, helping to weight the wheels and hold them steady in the turn. (If you lean or turn sharply with the inside pedal all the way down, it could strike the ground and at least throw you off balance, if not dumping you on the pavement. Always go into a turn with the outside pedal down and weighted like this.)

(Don't brake while you're in the turn... Feather your brakes to slow down enough to make your turn safely, release the brakes, and then make your turn. Braking, especially the front wheel, makes it want to go straight ahead, which is not want you want to do here.)

Remember what Oakleaf said -- your bike goes where you're looking -- Keep your head up, and your eyes focused on the line or direction you want to go. Imagine leading into the turn with your chin... that's where you want to go. Some folks want to point the inside knee into the turn as part of leaning into the direction you want to go; but press your outside thigh into the saddle, which pushes the top of the bike down and into the turn. At the same time, pull up a little bit on the bars with your outside hand. The bike will carve smoothly around the corner, leaning as much as it needs to, while you remain relatively upright, standing on the outside pedal.

To straighten out, relax your outside hand so you aren't pulling so much, and the bike will straighten out into the new direction.

Practice this a time or two or three until you feel comfortable, and the bike is carving smoothly thru the corners like you want it to. Now, go make a run thru your slalom course of cups/cans, passing alternately to the left and right of the cans. Repeat until you can cut smoothly thru the course.

Make each corner as shallow as the road and other traffic allows. The general rule is to approach a turn wide, cut close to the apex of the curve as straight as possible, and then exit wide. This has the effect of decreasing the angle of the turn, gives you greater stability, and lets you carry a little more speed and momentum into (and out of) the turn.

Take a little time every now and then to go "dancing with your bike" like this, learning how to balance and make the bike go where you want it to, and how far your balance will let you go before Mr. Gravity steps in too heavy-handedly...