The Earth spins and moves through space in an orbit around the sun. We don't "feel" this movement because we are spinning and moving through space with the Earth. Since we are standing on the Earth, we move as much (and as fast) as the Earth moves. We don't recognize this motion when we walk on the Earth's surface every day because we are always comparing our speed relative to other things that are also on our planet.

One way to recognize the Earth's movement is to look at objects that are not attached to Earth: like the sun or stars. The sun moves across the sky during the day because the Earth rotates (and we rotate with it). If you look at the stars at night, you'll notice that they rotate around the North Star (slowly)--which means we are rotating with the Earth and stars appear to move.

It is like being in a car on the freeway. When we are riding in a car, and the car is driving at a constant speed, we don't feel like anything IN the car is moving: our seat, our seatbelt, the windows, and all other things attached to the car. However, we (and all the things IN the car) are moving at the same speed as the car (down the freeway). We can notice that the car is moving if we look at objects NOT attached to the car: like plants next to the road or buildings we pass.

Likewise, we notice that the Earth moves through space when we look out at objects NOT on the Earth.

We certainly can - they're called earthquakes. However, most earthquakes are just too weak for us to feel.

We can't feel the Earth moving because we're moving at the same speed at the same time with our planet. Think about this: you're blindfolded in a car that's moving at a constant speed (no brakes and doesn't change directions). You won't feel it move, unless it changes speed (like braking, for example).

The same principle holds true in terms of Earth. You might say, "well, Earth doesn't go on a straight line, it goes in an ellipse." You are correct, but we don't feel this curve because this ellipse is so big that we don't 'feel' the curve but only a very small part of it at a time (which translates to a very short straight line). I hope I answered your question. Good luck with your studies!