All things on Earth move and rotate together with the Earth, therefore there are only a few physical effects on Earth which demonstrates its rotation (Coriolis forces, Foucault pendulum). This is elementary physics and astronomy, and you should read some textbooks or at least Wikipedia pages (start with https://en.wikipedia.org/wiki/Earth%27s_rotation). Also the Catholic Church has recognized this during the 20th century.
Sure. Even the effect of nutation and precession also affect the coordinate transformation of satellite orbit from the celestial frame to the terrestrial frame. There is also a special organization that deals with the measurement of earth rotation called IERS. You can learn more about it through this link https://www.iers.org/IERS/EN/Home/home_node.html
sorry for bad start but my IPad thinks it knows better! Wolfgang is spot on. However let us assume you are on a train travelling at 80mph. Everyone on the carriage is travelling at the same speed. If one passenger, a ballerina, gets up and pirouettes in the aisle, she is rotating and also moving forward at 80mph. The fact that all passengers, luggage, seats and tables are all moving at the speed of the train means that, unless you look out through the carriage window, you are not aware of the forward movement. Neither is the ballerina and she is rotating at the same time. So the Earth is rotating all the time while moving along its orbital course round the Sun, and along with the whole of our Solar System on a trajectory through space. But we are moving in the same way as the Earth so notice no movement. Only measurement of the Earth‘s position relative to the Sun, the Moon, our local planets and the stars shows the trajectories we and the Earth are travelling on.
Simply put, we know the sun is stationary therefore we know the earth rotates because we have night and day, times when we do not see the sun. We know the earth moves because of changes in seasons due to change in distance from the sun along its elliptical orbit. We are too small to feel the effects of rotation and movement of the earth because of the gravitational pull of the earth affecting us.
Dear Julian Simela: Please note that the seasons do not change due to the changes in distance from the sun, but due to the inclination of the Earth's axis relative to its orbital plane, from which follow different paths of the Sun's rays through the atmosphere during the year (or different heights of the sun as seen from a point on Earth). See https://en.wikipedia.org/wiki/Season#Axial_tilt and https://www.nationalgeographic.org/encyclopedia/season/ for more explanations. But anyway, the seasons are connected to the movement of the Earth around the Sun.
Note that the Sun is NOT stationary. It, along with the rest of its solar system, is moving through space at the same velocity as its planets and other satellites. Thus we on Earth are subject to several movements:
1. The solar system’s trajectory through space
2. The orbital trajectory around our Sun
3. The Earth’s rotation on its axis, which combined with our orbit gives us day and night and the seasons.
4. Our own personal movements as we walk, cycle, travel by car bus, train, plane across the surface of the earth.
There are minor adjustments that need to be compensated for depending on the scale we are working on, e.g. tidal fluctuations, the precession of the earth’s axis. All gets more complicated. We should be so thankful to be living in such an interesting universe, one we cannot fully fathom.
The solar system’s trajectory through space has also several components: The individual trajectory through the Galaxy, the movement together with the whole Galaxy (rotation + linear movement), and the expansion of the Universe (which means that as seen from any other galaxy our own Galaxy is moving away).