Earth rotates around its own axis which is tilted at 23.5 degrees, from the plane of its orbit around the sun, in the eastward direction. It takes almost 24 hours to complete one rotation. The Earth rotates at a speed of 460 m/s, that is approximately 1650 km/hr at the equator. The angular velocity of the Earth decreases as we keep moving toward the poles.
The Earth's rotational speed is slowing down with time. According to atomic clocks, the present day is 1.7 milliseconds slower than a century ago. It is not a big issue as it will take more than a trillion years for our Earth to stop rotating completely.
The Earth rotates at a great speed of 1600 km/hr near the equator. If the Earth suddenly started spinning in the opposite direction, everything not fixed to the ground would be launched at 3200 km/hr eastward. This would happen because everything will still be rotating due to inertia towards the east, while the ground suddenly moves towards the west.
This same forces will also cause giant tsunamis as the waters are also thrown towards east. The tsunamis are caused by the lack of gravity at the equator, thus forcing them to move towards the poles. While they move towards the poles they destroy everything in their path and change the whole structure of the planet. Without the Earth's rotation, the side facing the SUN would burn at high temperatures and the dark side would freeze.
Now, let us assume the case where the Earth rotates in westward direction since its birth. Now the sun rises from the west.
The above figure shows the flow of air around the globe due to Earth's rotation in the eastward direction. If the Earth spins in opposite direction, all these air currents are shown in the above figure also would flow opposite to their current path.
Oceanic currents and weather patterns are driven by Coriolis effects, which will reverse with the Earth's spin causing drastic changes in the weather patterns. This would cause major climatic changes. For example, the Sahara desert would be a wet place, California desert would also turn humid. On the other hand, the Florida swampland would dry out.
The present agricultural lands might be unfit for crops, heavy rainfall receiving areas may become dry areas whereas the deserts could receive a good amount of rainfall. The physics of launching of spaceships and satellites into space would also vary. All these changes can take place with the reversal of Earth's spin. Thus, change of spin direction can also cause major problems to our lifestyle and economy.
There will be effects from the change of the weather from the Coriolis Effect but how this could change the weather can only be estimated.