People don't feel the Earth spinning because they are spinning right along with it. Everything on Earth, including the atmosphere, is spinning at the same steady rate, so it feels as though nothing is moving at all. The effect is similar to being in a moving car. If the ride is steady, there is very little sensation of motion.
The Earth spins at a constant velocity of about 1,000 miles per hour. It also revolves around the sun at approximately 67,000 miles per hour. This motion goes unnoticed because the speed is constant and because everything on Earth is traveling in the same frame of reference. This is analogous to traveling on a jet. A jet might be flying at 500 miles per hour, but when a flight attendant pours tea, the tea goes from pot to cup instead of flying to the back of the plane. This is because the pot, the cup and the tea are all traveling at a constant speed in the same frame of reference.
While traveling at a constant velocity doesn't create the sensation of motion, acceleration and deceleration do. Speeding up a car pushes passengers back in their seats, and slamming on the brakes tosses everything inside the car forward, because the objects in the car retain their own forward momentum. However, if deceleration or acceleration is gradual enough, the effect is hardly noticeable.