You don't feel it because (1) it's small, and (2) because the gravity holding you to the Earth is about three hundred times as strong.

The Earth has a circumference of roughly 40,000 kilometers, which is traversed in a 24-hour day. This means that you are traveling at about 450 meters a second relative to the center of the Earth. I don't know how strong you are on algebra, but the physics equation that describes the Earth's motion is a=v²/r, where a is the acceleration toward the center of the planet that keeps you rotating around the Earth's surface instead of flying off on a tangent, v is the velocity, and r is the radius of the earth, 6400 kilometers.

Doing the calculations, we find that you're accelerating at 3.3 centimeters per second each second (that means that each second you are moving 3.3 centimeters per second faster. However, since the movement is going in the opposite direction at midnight, it equals out to zero over the whole of a day). You would feel this, though just barley.

However, the acceleration due to gravity is about three hundred times this, so even sitting on the equator (where the Earth's rotation is the fastest), it would be drowned out by gravity. You're probably wondering how you can be moving 450 meters per second (which is a little over a thousand miles per hour) and not feel it. That's because you don't feel motion - you feel changes in the rate of your motion. If you're sitting in a car going on a freeway, you don't notice any difference than if the car were going five miles per hour on a driveway. Likewise, you can't tell if you're riding in an airplane how fast the airplane is flying just by feel.