Why can't we feel the Earth moving?

Because you feel acceleration, not speed — and the Earth's motion is almost perfectly steady.

The speeds involved are enormous. At the equator, Earth's rotation carries you at roughly 1,670 km/h. Its orbit around the Sun moves you at about 107,000 km/h. The solar system orbits the galactic centre at around 828,000 km/h.

Why none of it is perceptible:

There is no such thing as absolute speed. Motion is only meaningful relative to something else, and everything around you — the ground, the air, the buildings — is moving identically. Relative to your surroundings, you are stationary.

Your senses detect change in motion. The vestibular system in your inner ear responds to acceleration and to changes in orientation, not to constant velocity. This is exactly why a smooth aircraft at 900 km/h feels like sitting still, and why you notice takeoff, turbulence and landing — those involve acceleration.

The accelerations that do exist are tiny. Rotation is a circular motion, so there is a centripetal acceleration — but spread over a 24-hour period and a very large radius, it amounts to roughly 0.03 m/s², about 0.3% of gravity. Orbital acceleration is smaller still. Both are far below the threshold of perception, and gravity swamps them.

The atmosphere moves with us, so there is no wind from rotation. This was a genuine historical objection to a moving Earth, resolved once it was understood that the air is carried along.

What we can detect with instruments:

The Coriolis effect — the deflection of moving air and water, which shapes weather systems. Note it does not determine which way a sink drains; that is decided by basin shape and initial motion.

A Foucault pendulum, whose swing plane appears to rotate, demonstrating Earth's rotation directly.

Weight varies slightly with latitude, being marginally lower at the equator.