It probably will not come as a surprise if told you that our planet, with its atmosphere and everything on it, is continuously spinning. The speed of Earth’s rotation at equator is approx. 1,675 kilometers per hour (1,040 mph). This means that at this very moment, you're roaming at something like 465 meters per second, or a little less if you're sited near to one of the poles. So the question here is, why can't we all feel it? Well there is simple physics behind all this. The precise answer lies in the nature of Earth's movement.
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Just think of this being in a fast moving train, when it's smoothly moving at a constant speed. While being in that train if you've unbuckled your seatbelt to get something or you just walk around, you can't feel the actual movement of the train. The reason behind all this is simple physics: you, the train, and everything else inside it is moving at the same at the same speed. In order to distinguish the movement of the train, you have to look at the something outside of the train like house, farms etc. It's the exactly same with Earth's rotation - our planet finishes a full turn around its axis every 23 hours and 56 minutes, spinning nonstop at nearly constant rate. One way to feel motion is to sense wind on your face - but remember that Earth's atmosphere is also travelling with us at the same exact speed.
If Earth were to alter its acceleration, we'd definitely feel that, and it wouldn't be pleasant at all, like a rapid slam on the brakes at a planetary scale (although the atmosphere would keep moving at the same exact 465 meters per second and completely wipe the surface of the planet). But just exactly like we can't sense the constant movement of a plane or a train, the spin of our massive space ride is generally unnoticeable, too.
As we are being precise here so, I would also like to mention that Earth is slowing down ever so slightly because of the Moon being a bit of a gravitational drag. It pulls on the tidal bulge of our planet, which bases tidal friction, pushing energy into the Moon's orbit.