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The Earth spins around its axis at 0.5 km/s as it orbits the Sun at 30 km/sec. Our entire Solar System slowly orbits around the Milky Way Galaxy at 250 km/sec. The Milky Way Galaxy belongs to the Local Group of galaxies, where it is also moving at 300 km/sec. 
If earth starts spinning fast, day-night will change it's time and proportion. Weather will severely change. May be you will get two summers and two winters every year. In a word, local properties of earth will change. You can see and detect these changes. 
If earth starts orbiting the Sun faster, then it can not do that by going in the same orbit. It will change orbit and move away from the Sun to settle in dynamic equilibrium. To do that moon will change it's apogee and perigee as well. You can find this change seeing at different starts at night sky, starts will appear at different angles. We would also have to look at the speed of winds. here watch this video for more info